9722:
9274:
2049:
11711:. The vector itself does not change when the basis is transformed; instead, the components of the vector make a change that cancels the change in the basis. In other words, if the reference axes (and the basis derived from it) were rotated in one direction, the component representation of the vector would rotate in the opposite way to generate the same final vector. Similarly, if the reference axes were stretched in one direction, the components of the vector would reduce in an exactly compensating way. Mathematically, if the basis undergoes a transformation described by an
8756:
13063:
5591:
5388:
10553:
8401:
9717:{\displaystyle {\begin{aligned}c_{11}&=\mathbf {n} _{1}\cdot \mathbf {e} _{1}\\c_{12}&=\mathbf {n} _{1}\cdot \mathbf {e} _{2}\\c_{13}&=\mathbf {n} _{1}\cdot \mathbf {e} _{3}\\c_{21}&=\mathbf {n} _{2}\cdot \mathbf {e} _{1}\\c_{22}&=\mathbf {n} _{2}\cdot \mathbf {e} _{2}\\c_{23}&=\mathbf {n} _{2}\cdot \mathbf {e} _{3}\\c_{31}&=\mathbf {n} _{3}\cdot \mathbf {e} _{1}\\c_{32}&=\mathbf {n} _{3}\cdot \mathbf {e} _{2}\\c_{33}&=\mathbf {n} _{3}\cdot \mathbf {e} _{3}\end{aligned}}}
5362:
1728:
3596:
5028:
6688:
10109:
5912:
13327:
42:
8392:
11566:
2072:
8751:{\displaystyle {\begin{aligned}u&=p\mathbf {e} _{1}\cdot \mathbf {n} _{1}+q\mathbf {e} _{2}\cdot \mathbf {n} _{1}+r\mathbf {e} _{3}\cdot \mathbf {n} _{1},\\v&=p\mathbf {e} _{1}\cdot \mathbf {n} _{2}+q\mathbf {e} _{2}\cdot \mathbf {n} _{2}+r\mathbf {e} _{3}\cdot \mathbf {n} _{2},\\w&=p\mathbf {e} _{1}\cdot \mathbf {n} _{3}+q\mathbf {e} _{2}\cdot \mathbf {n} _{3}+r\mathbf {e} _{3}\cdot \mathbf {n} _{3}.\end{aligned}}}
11525:
7529:
6159:
2084:
11632:
8109:
10548:{\displaystyle {\begin{aligned}(a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}+a_{4}{\mathbf {e} }_{4})&+(b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}+b_{4}{\mathbf {e} }_{4})=\\(a_{1}+b_{1}){\mathbf {e} }_{1}+(a_{2}+b_{2}){\mathbf {e} }_{2}&+(a_{3}+b_{3}){\mathbf {e} }_{3}+(a_{4}+b_{4}){\mathbf {e} }_{4}.\end{aligned}}}
1670:, depending on how the transformation of the vector's components is related to the transformation of the basis. In general, contravariant vectors are "regular vectors" with units of distance (such as a displacement), or distance times some other unit (such as velocity or acceleration); covariant vectors, on the other hand, have units of one-over-distance such as
7328:
5984:
9186:
10104:
10779:
6990:
8043:
7790:
7321:
8387:{\displaystyle {\begin{aligned}u&=(p\mathbf {e} _{1}+q\mathbf {e} _{2}+r\mathbf {e} _{3})\cdot \mathbf {n} _{1},\\v&=(p\mathbf {e} _{1}+q\mathbf {e} _{2}+r\mathbf {e} _{3})\cdot \mathbf {n} _{2},\\w&=(p\mathbf {e} _{1}+q\mathbf {e} _{2}+r\mathbf {e} _{3})\cdot \mathbf {n} _{3}.\end{aligned}}}
11804:
is a contravariant vector: if the coordinates of space are stretched, rotated, or twisted, then the components of the velocity transform in the same way. On the other hand, for instance, a triple consisting of the length, width, and height of a rectangular box could make up the three components of an
1583:
of two vectors is defined—a scalar-valued product of two vectors—then it is also possible to define a length; the dot product gives a convenient algebraic characterization of both angle (a function of the dot product between any two non-zero vectors) and length (the square root of the dot product of
519:. In this context, vectors are abstract entities which may or may not be characterized by a magnitude and a direction. This generalized definition implies that the above-mentioned geometric entities are a special kind of abstract vectors, as they are elements of a special kind of vector space called
1662:. When the basis is transformed, for example by rotation or stretching, then the components of any vector in terms of that basis also transform in an opposite sense. The vector itself has not changed, but the basis has, so the components of the vector must change to compensate. The vector is called
8075:
in both cases. It is common to encounter vectors known in terms of different bases (for example, one basis fixed to the Earth and a second basis fixed to a moving vehicle). In such a case it is necessary to develop a method to convert between bases so the basic vector operations such as addition
8976:
10820:
length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 newton" and "5 m" are both represented with an arrow of 2 cm, the scales are 1 m:50 N and 1:250 respectively. Equal length of vectors of
2063:
of the diagram is sometimes desired. These vectors are commonly shown as small circles. A circle with a dot at its centre (Unicode U+2299 ⊙) indicates a vector pointing out of the front of the diagram, toward the viewer. A circle with a cross inscribed in it (Unicode U+2297 ⊗) indicates a vector
3149:
8967:
11250:
3537:
7524:{\displaystyle (\mathbf {a} \ \mathbf {b} \ \mathbf {c} )=(\mathbf {c} \ \mathbf {a} \ \mathbf {b} )=(\mathbf {b} \ \mathbf {c} \ \mathbf {a} )=-(\mathbf {a} \ \mathbf {c} \ \mathbf {b} )=-(\mathbf {b} \ \mathbf {a} \ \mathbf {c} )=-(\mathbf {c} \ \mathbf {b} \ \mathbf {a} ).}
5298:
4983:
388:
The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the
6628:
6154:{\displaystyle \mathbf {\hat {a}} ={\frac {\mathbf {a} }{\left\|\mathbf {a} \right\|}}={\frac {a_{1}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{1}+{\frac {a_{2}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{2}+{\frac {a_{3}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{3}}
9880:
1713:. The vectors described in this article are a very special case of this general definition, because they are contravariant with respect to the ambient space. Contravariance captures the physical intuition behind the idea that a vector has "magnitude and direction".
10571:
6760:
7916:
7663:
6352:
7153:
7111:
417:(Theory of the Ebb and Flow) was the first system of spatial analysis that is similar to today's system, and had ideas corresponding to the cross product, scalar product and vector differentiation. Grassmann's work was largely neglected until the 1870s.
5552:
11014:
4123:
2639:
4669:
4560:
4340:
4235:
2795:
4011:
3793:
3003:
3730:
2996:
523:. This particular article is about vectors strictly defined as arrows in Euclidean space. When it becomes necessary to distinguish these special vectors from vectors as defined in pure mathematics, they are sometimes referred to as
6215:, or simply 0. Unlike any other vector, it has an arbitrary or indeterminate direction, and cannot be normalized (that is, there is no unit vector that is a multiple of the zero vector). The sum of the zero vector with any vector
8765:
11080:
5902:
5813:
191:
means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of
7650:
3412:
7903:
981:
Sometimes, Euclidean vectors are considered without reference to a
Euclidean space. In this case, a Euclidean vector is an element of a normed vector space of finite dimension over the reals, or, typically, an element of the
6501:
1425:
This coordinate representation of free vectors allows their algebraic features to be expressed in a convenient numerical fashion. For example, the sum of the two (free) vectors (1, 2, 3) and (−2, 0, 4) is the (free) vector
11510:
5138:
4823:
3663:
3323:
6553:
2491:
11348:
9181:{\displaystyle {\begin{bmatrix}u\\v\\w\\\end{bmatrix}}={\begin{bmatrix}c_{11}&c_{12}&c_{13}\\c_{21}&c_{22}&c_{23}\\c_{31}&c_{32}&c_{33}\end{bmatrix}}{\begin{bmatrix}p\\q\\r\end{bmatrix}}.}
4780:
4438:
7123:, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors
661:
1562:
6286:
1418:. It is then determined by the coordinates of that bound vector's terminal point. Thus the free vector represented by (1, 0, 0) is a vector of unit length—pointing along the direction of the positive
7039:
5936:
is any vector with a length of one; normally unit vectors are used simply to indicate direction. A vector of arbitrary length can be divided by its length to create a unit vector. This is known as
3264:
614:
299:
The vector concept, as it is known today, is the result of a gradual development over a period of more than 200 years. About a dozen people contributed significantly to its development. In 1835,
10114:
9279:
8770:
8406:
8114:
7921:
7668:
1950:
1328:. Examples of quantities that have magnitude and direction, but fail to follow the rules of vector addition, are angular displacement and electric current. Consequently, these are not vectors.
1280:
Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has magnitude, has direction, and which adheres to the rules of vector addition. An example is
2371:
2301:
5427:
175:
11428:
10911:
4020:
2510:
267:
acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example,
4569:
4460:
4240:
4135:
2654:
3873:
1898:
447:
of two vectors from the complete quaternion product. This approach made vector calculations available to engineers—and others working in three dimensions and skeptical of the fourth.
1187:
809:
2218:
1678:) from meters to millimeters, a scale factor of 1/1000, a displacement of 1 m becomes 1000 mm—a contravariant change in numerical value. In contrast, a gradient of 1
1384:
933:
895:
10099:{\displaystyle (a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2})+(b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2})=(a_{1}+b_{1}){\mathbf {e} }_{1}+(a_{2}+b_{2}){\mathbf {e} }_{2},}
9877:
With the exception of the cross and triple products, the above formulae generalise to two dimensions and higher dimensions. For example, addition generalises to two dimensions as
972:
9869:
By applying several matrix multiplications in succession, any vector can be expressed in any basis so long as the set of direction cosines is known relating the successive bases.
10774:{\displaystyle (a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2})\wedge (b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2})=(a_{1}b_{2}-a_{2}b_{1})\mathbf {e} _{1}\mathbf {e} _{2}.}
6985:{\displaystyle {\mathbf {a} }\times {\mathbf {b} }=(a_{2}b_{3}-a_{3}b_{2}){\mathbf {e} }_{1}+(a_{3}b_{1}-a_{1}b_{3}){\mathbf {e} }_{2}+(a_{1}b_{2}-a_{2}b_{1}){\mathbf {e} }_{3}.}
2184:
3353:
10825:
inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance.
10787:
is similar to the cross product in that its result is a vector orthogonal to the two arguments; there is however no natural way of selecting one of the possible such products.
8038:{\displaystyle {\begin{aligned}u&=\mathbf {a} \cdot \mathbf {n} _{1},\\v&=\mathbf {a} \cdot \mathbf {n} _{2},\\w&=\mathbf {a} \cdot \mathbf {n} _{3}.\end{aligned}}}
7785:{\displaystyle {\begin{aligned}p&=\mathbf {a} \cdot \mathbf {e} _{1},\\q&=\mathbf {a} \cdot \mathbf {e} _{2},\\r&=\mathbf {a} \cdot \mathbf {e} _{3}.\end{aligned}}}
2826:
1219:
1128:
1077:
1045:
1012:
1988:
7563:}. However, a vector can be expressed in terms of any number of different bases that are not necessarily aligned with each other, and still remain the same vector. In the
5854:
1812:
1789:
1766:
978:. It has been proven that the two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations.
836:). In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to
7316:{\displaystyle (\mathbf {a} \ \mathbf {b} \ \mathbf {c} )={\begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{vmatrix}}}
5727:
7576:
6209:
1855:
1411:
In
Cartesian coordinates, a free vector may be thought of in terms of a corresponding bound vector, in this sense, whose initial point has the coordinates of the origin
7832:
275:), their magnitude and direction can still be represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the
6404:
1300:, since it has a magnitude and direction and follows the rules of vector addition. Vectors also describe many other physical quantities, such as linear displacement,
12432:
3739:
11809:, but this vector would not be contravariant, since rotating the box does not change the box's length, width, and height. Examples of contravariant vectors include
11774:
11442:
6532:
dimensions. The cross product differs from the dot product primarily in that the result of the cross product of two vectors is a vector. The cross product, denoted
3676:
7024:) is not really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is sometimes denoted by (
6994:
For arbitrary choices of spatial orientation (that is, allowing for left-handed as well as right-handed coordinate systems) the cross product of two vectors is a
4132:
Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors
2388:
11289:
4678:
1146:
of the associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines
3603:
The decomposition or resolution of a vector into components is not unique, because it depends on the choice of the axes on which the vector is projected.
11699:
An alternative characterization of
Euclidean vectors, especially in physics, describes them as lists of quantities which behave in a certain way under a
558:. When only the magnitude and direction of the vector matter, and the particular initial or terminal points are of no importance, the vector is called a
12514:
4345:
3144:{\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}+\mathbf {a} _{3}=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3},}
11780:. This important requirement is what distinguishes a contravariant vector from any other triple of physically meaningful quantities. For example, if
9866:
to relate the two vector bases, so the basis conversions can be performed directly, without having to work out all the dot products described above.
1600:
defined by two vectors (used as sides of the parallelogram). In any dimension (and, in particular, higher dimensions), it is possible to define the
11650:
11910:
has an angular velocity vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, the
11840:, the requirement that the components of a vector transform according to the same matrix of the coordinate transition is equivalent to defining a
3609:
3269:
1429:
8962:{\displaystyle {\begin{aligned}u&=c_{11}p+c_{12}q+c_{13}r,\\v&=c_{21}p+c_{22}q+c_{23}r,\\w&=c_{31}p+c_{32}q+c_{33}r,\end{aligned}}}
11867:
gain a minus sign. A transformation that switches right-handedness to left-handedness and vice versa like a mirror does is said to change the
11245:{\displaystyle {\mathbf {y} }-{\mathbf {x} }=(y_{1}-x_{1}){\mathbf {e} }_{1}+(y_{2}-x_{2}){\mathbf {e} }_{2}+(y_{3}-x_{3}){\mathbf {e} }_{3}.}
3669:
in which to represent a vector is not mandated. Vectors can also be expressed in terms of an arbitrary basis, including the unit vectors of a
1634:
However, it is not always possible or desirable to define the length of a vector. This more general type of spatial vector is the subject of
3532:{\displaystyle \mathbf {a} =\mathbf {a} _{x}+\mathbf {a} _{y}+\mathbf {a} _{z}=a_{x}{\mathbf {i} }+a_{y}{\mathbf {j} }+a_{z}{\mathbf {k} }.}
12879:
3805:
as a function of time or space. For example, a vector in three-dimensional space can be decomposed with respect to two axes, respectively
11918:
angular velocity vector of the wheel still points to the left, corresponding to the minus sign. Other examples of pseudovectors include
8104:
containing the information that relates the two bases. Such an expression can be formed by substitution of the above equations to form
13376:
2306:
2236:
7119:
which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are
1264:
is −15 N. In either case, the magnitude of the vector is 15 N. Likewise, the vector representation of a displacement Δ
2188:
The notion that the tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation
11396:
5293:{\displaystyle \mathbf {a} -\mathbf {b} =(a_{1}-b_{1})\mathbf {e} _{1}+(a_{2}-b_{2})\mathbf {e} _{2}+(a_{3}-b_{3})\mathbf {e} _{3}.}
4978:{\displaystyle \mathbf {a} +\mathbf {b} =(a_{1}+b_{1})\mathbf {e} _{1}+(a_{2}+b_{2})\mathbf {e} _{2}+(a_{3}+b_{3})\mathbf {e} _{3}.}
12921:
11972:
11849:
1683:
3795:). The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry, respectively.
6623:{\displaystyle \mathbf {a} \times \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\sin(\theta )\,\mathbf {n} }
3844:
In these cases, each of the components may be in turn decomposed with respect to a fixed coordinate system or basis set (e.g., a
13254:
13312:
1340:, a bound vector can be represented by identifying the coordinates of its initial and terminal point. For instance, the points
679:, then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin.
12472:
Vector
Analysis: A Text-book for the Use of Students of Mathematics and Physics, Founded upon the Lectures of J. Willard Gibbs
12845:
12821:
12716:
12672:
12646:
9809:" (because it can be imagined as the "rotation" of a vector from one basis to another), or the "direction cosine matrix from
1018:. This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is,
7142:), if the three vectors are thought of as rows (or columns, but in the same order), the scalar triple product is simply the
3231:
619:
1575:
or magnitude and a direction to vectors. In addition, the notion of direction is strictly associated with the notion of an
582:
12326:
2146:
1919:
410:
813:
A Euclidean vector is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the
12793:
12767:
12693:
12599:
12555:
12212:
11686:
11668:
11613:
11587:
11552:
11538:
141:
11595:
5361:
13302:
12035:
11863:
Some vectors transform like contravariant vectors, except that when they are reflected through a mirror, they flip
315:
on the pairs of points (bipoints) in the plane, and thus erected the first space of vectors in the plane. The term
31:
6347:{\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta ,}
1272:
would be 4 m or −4 m, depending on its direction, and its magnitude would be 4 m regardless.
475:, adapted from Gibbs's lectures, which banished any mention of quaternions in the development of vector calculus.
13264:
13200:
12835:
10839:
Often in areas of physics and mathematics, a vector evolves in time, meaning that it depends on a time parameter
10784:
6529:
17:
1604:, which (among other things) supplies an algebraic characterization of the area and orientation in space of the
11591:
7106:{\displaystyle (\mathbf {a} \ \mathbf {b} \ \mathbf {c} )=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ).}
1870:
1727:
393:
Several other mathematicians developed vector-like systems in the middle of the nineteenth century, including
9858:
The advantage of this method is that a direction cosine matrix can usually be obtained independently by using
5027:
12872:
12737:
12518:
4452:
3670:
1157:
779:
2191:
1646:, where many quantities of interest can be considered vectors in a space with no notion of length or angle.
1357:
908:
870:
13371:
13042:
12914:
12321:
8071:
relate to the unit vectors in such a way that the resulting vector sum is exactly the same physical vector
950:
686:
also has generalizations to higher dimensions, and to more formal approaches with much wider applications.
11929:
This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying
3584:
of the vector on a set of mutually perpendicular reference axes (basis vectors). The vector is said to be
1047:
is a
Euclidean space, with itself as an associated vector space, and the dot product as an inner product.
13386:
13147:
12997:
12867:
12732:
9236:
5572:
copies of the vector in a line where the endpoint of one vector is the initial point of the next vector.
5547:{\displaystyle r\mathbf {a} =(ra_{1})\mathbf {e} _{1}+(ra_{2})\mathbf {e} _{2}+(ra_{3})\mathbf {e} _{3}.}
3867:
3798:
The choice of a basis does not affect the properties of a vector or its behaviour under transformations.
3733:
2810:
2119:
2099:
1642:(for bound vectors, as each represented by an ordered pair of "points"). One physical example comes from
1337:
1253:
663:
in space represent the same free vector if they have the same magnitude and direction: that is, they are
3331:
13381:
13052:
12946:
11009:{\displaystyle {\mathbf {x} }=x_{1}{\mathbf {e} }_{1}+x_{2}{\mathbf {e} }_{2}+x_{3}{\mathbf {e} }_{3}.}
4118:{\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}.}
2634:{\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\\\end{bmatrix}}=^{\operatorname {T} }.}
1244:
has a direction and a magnitude, it may be seen as a vector. As an example, consider a rightward force
1192:
1101:
898:
10558:
4664:{\displaystyle {\mathbf {b} }=b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}}
4555:{\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}}
4335:{\displaystyle {\mathbf {b} }=b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}}
4230:{\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}}
2090:
In order to calculate with vectors, the graphical representation may be too cumbersome. Vectors in an
1053:
1021:
988:
13356:
13292:
12941:
12163:
7007:
2790:{\displaystyle {\mathbf {e} }_{1}=(1,0,0),\ {\mathbf {e} }_{2}=(0,1,0),\ {\mathbf {e} }_{3}=(0,0,1).}
1969:
1571:
In the geometrical and physical settings, it is sometimes possible to associate, in a natural way, a
12889:
10867:
by differentiating or integrating the components of the vector, and many of the familiar rules from
7325:
The scalar triple product is linear in all three entries and anti-symmetric in the following sense:
4006:{\displaystyle {\mathbf {e} }_{1}=(1,0,0),\ {\mathbf {e} }_{2}=(0,1,0),\ {\mathbf {e} }_{3}=(0,0,1)}
13361:
13284:
13167:
12883:
11576:
10822:
6399:
The dot product can also be defined as the sum of the products of the components of each vector as
5579:
is negative, then the vector changes direction: it flips around by an angle of 180°. Two examples (
3849:
1139:
1083:
436:
9817:" (because it contains direction cosines). The properties of a rotation matrix are such that its
1795:
1772:
1749:
13366:
13330:
13259:
13037:
12907:
12168:
12153:
11580:
8101:
5683:
2042:
707:
664:
492:
465:, published in 1881, presents what is essentially the modern system of vector analysis. In 1901,
304:
244:
vectors as an example of the more generalized concept of vectors defined simply as elements of a
193:
85:
35:
7538:
All examples thus far have dealt with vectors expressed in terms of the same basis, namely, the
6185:
5418:) to distinguish them from vectors. The operation of multiplying a vector by a scalar is called
1831:
13094:
13027:
13017:
12066:
11708:
11390:
10834:
6373:
3802:
3788:{\displaystyle \mathbf {\hat {r}} ,{\boldsymbol {\hat {\theta }}},{\boldsymbol {\hat {\phi }}}}
3666:
3567:
1620:
1593:
975:
320:
12545:
11873:
of space. A vector which gains a minus sign when the orientation of space changes is called a
10557:
The cross product does not readily generalise to other dimensions, though the closely related
564:. The distinction between bound and free vectors is especially relevant in mechanics, where a
13109:
13104:
13099:
13032:
12977:
12176:
11837:
11810:
11544:
11437:
11074:
9786:
7115:
It has three primary uses. First, the absolute value of the box product is the volume of the
5382:
3725:{\displaystyle {\boldsymbol {\hat {\rho }}},{\boldsymbol {\hat {\phi }}},\mathbf {\hat {z}} }
1905:
1722:
1609:
1301:
1260:
is represented by the vector 15 N, and if positive points leftward, then the vector for
1147:
983:
500:
406:
272:
120:
115:
12862:
12727:
11980:, a non-Euclidean vector in Minkowski space (i.e. four-dimensional spacetime), important in
3861:
2048:
13119:
13084:
13071:
12962:
12467:
11981:
11750:
10817:
10810:
7147:
5074:
are bound vectors that have the same base point, this point will also be the base point of
3819:
3801:
A vector can also be broken up with respect to "non-fixed" basis vectors that change their
3555:
2504:
2115:
1825:
1305:
837:
676:
512:
466:
450:
312:
111:
93:
11646:
4794:) if they have the same direction but not necessarily the same magnitude. Two vectors are
2991:{\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3})=a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1),\ }
1900:, which is a convention for indicating boldface type. If the vector represents a directed
8:
13351:
13297:
13177:
13152:
13002:
12893:
12020:
11869:
10856:
10813:
7120:
6265:
5818:
3834:
2060:
1706:
864:
454:
431:
308:
280:
205:
12504:
respectively. This resolution may be accomplished by constructing the parallelogram ..."
12231:
9854:
the rows and columns are orthogonal unit vectors, therefore their dot products are zero.
1623:, a vector's squared length can be positive, negative, or zero. An important example is
13007:
12782:
12139:
11707:
is required to have components that "transform opposite to the basis" under changes of
5689:
2064:
pointing into and behind the diagram. These can be thought of as viewing the tip of an
1628:
1324:, are represented as a system of vectors at each point of a physical space; that is, a
699:
695:
573:
496:
461:, separated off their vector part for independent treatment. The first half of Gibbs's
418:
228:
have close analogues for vectors, operations which obey the familiar algebraic laws of
13205:
13162:
13089:
12982:
12841:
12817:
12811:
12789:
12763:
12712:
12689:
12668:
12642:
12605:
12595:
12551:
12208:
11967:
11712:
11700:
5897:{\displaystyle \left\|\mathbf {a} \right\|={\sqrt {\mathbf {a} \cdot \mathbf {a} }}.}
3581:
2095:
1710:
1229:
1143:
1135:
940:
703:
398:
359:
300:
276:
268:
107:
12667:. Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications. Wiley.
12641:. Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra. Wiley.
5808:{\displaystyle \left\|\mathbf {a} \right\|={\sqrt {a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}},}
5590:
5302:
Subtraction of two vectors can be geometrically illustrated as follows: to subtract
4798:
if they have the same or opposite direction but not necessarily the same magnitude.
13210:
13114:
12317:
12015:
11957:
11947:
11899:
11891:
to distinguish them from pseudovectors. Pseudovectors occur most frequently as the
9822:
7645:{\displaystyle \mathbf {a} =p\mathbf {e} _{1}+q\mathbf {e} _{2}+r\mathbf {e} _{3}.}
3222:
3176:
2124:
1601:
1313:
508:
225:
12262:
7898:{\displaystyle \mathbf {a} =u\mathbf {n} _{1}+v\mathbf {n} _{2}+w\mathbf {n} _{3}}
2797:
These have the intuitive interpretation as vectors of unit length pointing up the
2045:, while the direction in which the arrow points indicates the vector's direction.
402:
13269:
13062:
13022:
13012:
12831:
12594:(3rd ed.). Reston, Va.: American Institute of Aeronautics and Astronautics.
12061:
12056:
10905:
10859:
representation of the trajectory of the particle. Vector-valued functions can be
10796:
9798:
6732:
6661:
5407:
5387:
4786:
1959:
1675:
1624:
1309:
1241:
569:
520:
504:
471:
394:
241:
102:
6496:{\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}.}
3228:
In introductory physics textbooks, the standard basis vectors are often denoted
1658:
of components, or list of numbers, that act as scalar coefficients for a set of
13274:
13195:
12930:
12456:
11986:
11962:
11919:
11818:
9818:
7116:
5718:
5706:
5610:
3551:
2648:
2065:
1643:
1584:
a vector by itself). In three dimensions, it is further possible to define the
1321:
1317:
902:
856:
367:
363:
237:
221:
89:
12333:
11505:{\displaystyle W={\mathbf {F} }\cdot ({\mathbf {x} }_{2}-{\mathbf {x} }_{1}).}
4013:
and assumes that all vectors have the origin as a common base point. A vector
1690:
are another type of quantity that behave in this way; a vector is one type of
311:
line segments of the same length and orientation. Essentially, he realized an
283:
and transform in a similar way under changes of the coordinate system include
13345:
13307:
13230:
13190:
13157:
13137:
12683:
12661:
12636:
12609:
12086:"Can be brought to the same straight line by means of parallel displacement".
12051:
12005:
12000:
11892:
11853:
10871:
continue to hold for the derivative and integral of vector-valued functions.
6649:
6511:
5055:
5000:
The addition may be represented graphically by placing the tail of the arrow
3838:
2496:
2056:
1597:
1585:
1292:
could be represented by the vector (0, 5) (in 2 dimensions with the positive
1249:
757:
672:
444:
233:
229:
12287:
5391:
Scalar multiplication of a vector by a factor of 3 stretches the vector out.
13240:
13129:
13079:
12972:
12777:
12025:
11952:
11943:
11875:
11830:
11806:
11368:
10566:
9859:
6995:
4807:
3658:{\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} }
3359:). In this case, the scalar and vector components are denoted respectively
3318:{\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} }
3188:
on the basis vectors or, equivalently, on the corresponding
Cartesian axes
1994:
1702:
1659:
1639:
1635:
1325:
1269:
944:
849:
814:
711:
516:
284:
260:
245:
97:
6178:
is the vector with length zero. Written out in coordinates, the vector is
3599:
Illustration of tangential and normal components of a vector to a surface.
1993:
Vectors are usually shown in graphs or other diagrams as arrows (directed
491:, a vector is typically regarded as a geometric entity characterized by a
13220:
13185:
13142:
12987:
12656:
12632:
12046:
12010:
11977:
7143:
6664:
system. The right-handedness constraint is necessary because there exist
6246:
6169:
5927:
5846:
5400:
3595:
3356:
2135:
As an example in two dimensions (see figure), the vector from the origin
1698:
1580:
1095:
1015:
936:
488:
440:
342:
217:
209:
65:
57:
11856:, and the rules for transforming a contravariant vector follow from the
10821:
different dimension has no particular significance unless there is some
1654:
In physics, as well as mathematics, a vector is often identified with a
1090:. This is motivated by the fact that every Euclidean space of dimension
124:. A vector is frequently depicted graphically as an arrow connecting an
13249:
12992:
12755:
12474:, by E.B. Wilson, Chares Scribner's Sons, New York, p. 15: "Any vector
12030:
11857:
10860:
9863:
6728:
6687:
5911:
3326:
2500:
2486:{\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3},\cdots ,a_{n-1},a_{n}).}
1130:
More precisely, given such a
Euclidean space, one may choose any point
324:
11343:{\displaystyle {\mathbf {x} }_{t}=t{\mathbf {v} }+{\mathbf {x} }_{0},}
1682:/m becomes 0.001 K/mm—a covariant change in value (for more, see
1189:
These choices define an isomorphism of the given
Euclidean space onto
13047:
12708:
11991:
4775:{\displaystyle a_{1}=-b_{1},\quad a_{2}=-b_{2},\quad a_{3}=-b_{3}.\,}
3558:
commonly used in higher level mathematics, physics, and engineering.
2651:
vectors. For instance, in three dimensions, there are three of them:
2102:. The endpoint of a vector can be identified with an ordered list of
1962:
literature, it was especially common to represent vectors with small
1821:
1743:
371:
41:
11565:
9265:
of the angle between two unit vectors, which is also equal to their
13215:
12184:
11930:
11852:
rank one. Alternatively, a contravariant vector is defined to be a
11822:
11814:
11797:
11272:
10868:
10864:
10562:
8971:
and these equations can be expressed as the single matrix equation
5564:. Geometrically, this can be visualized (at least in the case when
3829:
1901:
1671:
1281:
841:
256:
213:
11260:. The length of this vector gives the straight-line distance from
4433:{\displaystyle a_{1}=b_{1},\quad a_{2}=b_{2},\quad a_{3}=b_{3}.\,}
2071:
12899:
3576:, a vector is often described by a set of vector components that
2075:
A vector in the
Cartesian plane, showing the position of a point
1963:
484:
252:
61:
10809:
In abstract vector spaces, the length of the arrow depends on a
358:. Like Bellavitis, Hamilton viewed vectors as representative of
307:. Working in a Euclidean plane, he made equipollent any pair of
13225:
12041:
11923:
11845:
11017:
9262:
5677:
4442:
3825:
1691:
1687:
1679:
764:, more precisely, a Euclidean vector. The equivalence class of
288:
11514:
7533:
6376:
for an explanation of cosine). Geometrically, this means that
3580:
to form the given vector. Typically, these components are the
11926:, or more generally any cross product of two (true) vectors.
11914:
of this angular velocity vector points to the right, but the
11907:
11826:
11433:
11389:
is a vector with dimensions of mass×length/time (N m s ) and
11386:
11279:
6361:
2233:), vectors are identified with triples of scalar components:
2220:
is usually deemed not necessary (and is indeed rarely used).
2111:
1864:
1655:
1297:
1285:
1222:
1154:
of the space, as the coordinates on this basis of the vector
845:
565:
303:
abstracted the basic idea when he established the concept of
279:
used to describe it. Other vector-like objects that describe
264:
12515:"U. Guelph Physics Dept., "Torque and Angular Acceleration""
10816:. If it represents, for example, a force, the "scale" is of
6384:
are drawn with a common start point, and then the length of
5940:
a vector. A unit vector is often indicated with a hat as in
2083:
1296:-axis as 'up'). Another quantity represented by a vector is
855:
In modern geometry, Euclidean spaces are often defined from
439:. Clifford simplified the quaternion study by isolating the
12813:
Vectors, Tensors and the Basic
Equations of Fluid Mechanics
6746:
can be interpreted as the area of the parallelogram having
5318:
at the same point, and then draw an arrow from the head of
2068:
head on and viewing the flights of an arrow from the back.
1589:
2041:. The length of the arrow is proportional to the vector's
1557:{\displaystyle (1,2,3)+(-2,0,4)=(1-2,2+0,3+4)=(-1,2,7)\,.}
552:; such a condition may be emphasized calling the result a
12185:"Earliest Known Uses of Some of the Words of Mathematics"
11903:
11278:
of a point or particle is a vector, its length gives the
426:
421:
carried the quaternion standard after Hamilton. His 1867
702:), vectors were introduced (during the 19th century) as
12685:
Introduction to Tensor Calculus and Continuum Mechanics
2118:
of the endpoint of the vector, with respect to a given
1867:(~) or a wavy underline drawn beneath the symbol, e.g.
1863:, especially in handwriting. Alternatively, some use a
10874:
10801:
Vectors have many uses in physics and other sciences.
9190:
This matrix equation relates the scalar components of
9147:
9021:
8985:
8760:
Replacing each dot product with a unique scalar gives
8076:
and subtraction can be performed. One way to express
7192:
3259:{\displaystyle \mathbf {i} ,\mathbf {j} ,\mathbf {k} }
2527:
2132:) of the vector on the axes of the coordinate system.
1588:, which supplies an algebraic characterization of the
1228:
of its Cartesian coordinates, and every vector to its
675:. If the Euclidean space is equipped with a choice of
656:{\displaystyle {\stackrel {\,\longrightarrow }{A'B'}}}
144:
96:. Euclidean vectors can be added and scaled to form a
12359:
London, Edinburgh & Dublin Philosophical Magazine
12146:= "I carry". For historical development of the word
11753:
11445:
11399:
11292:
11083:
10914:
10904:) in three-dimensional space can be represented as a
10574:
10112:
9883:
9277:
8979:
8768:
8404:
8112:
7919:
7835:
7666:
7579:
7331:
7156:
7042:
6763:
6556:
6407:
6289:
6188:
5987:
5857:
5730:
5430:
5141:
4826:
4681:
4572:
4463:
4348:
4243:
4138:
4023:
3876:
3742:
3679:
3612:
3415:
3334:
3272:
3234:
3006:
2829:
2657:
2513:
2391:
2309:
2239:
2194:
2149:
1972:
1922:
1873:
1834:
1798:
1775:
1752:
1432:
1360:
1195:
1160:
1104:
1056:
1024:
991:
953:
911:
873:
782:
622:
585:
12830:
12592:
Applied mathematics in integrated navigation systems
9844:
The properties of a direction cosine matrix, C are:
5975:, scale the vector by the reciprocal of its length ‖
3606:
Moreover, the use of Cartesian unit vectors such as
609:{\displaystyle {\stackrel {\,\longrightarrow }{AB}}}
12151:
11641:
may be too technical for most readers to understand
10847:represents the position vector of a particle, then
5845:This happens to be equal to the square root of the
2094:-dimensional Euclidean space can be represented as
947:for details of this construction). The elements of
12781:
12660:
12152:
11768:
11504:
11422:
11342:
11244:
11008:
10773:
10547:
10098:
9716:
9180:
8961:
8750:
8386:
8037:
7897:
7784:
7644:
7523:
7315:
7105:
6984:
6622:
6495:
6388:is multiplied with the length of the component of
6346:
6203:
6153:
5896:
5807:
5546:
5292:
4977:
4774:
4663:
4554:
4432:
4334:
4229:
4117:
4005:
3787:
3724:
3657:
3531:
3347:
3317:
3258:
3143:
2990:
2789:
2633:
2485:
2365:
2295:
2212:
2178:
1982:
1945:{\displaystyle {\stackrel {\longrightarrow }{AB}}}
1944:
1892:
1849:
1806:
1783:
1760:
1556:
1378:
1213:
1181:
1122:
1071:
1039:
1006:
966:
927:
889:
803:
655:
608:
169:
12547:Handbook of mathematics and computational science
5994:
3779:
3764:
3749:
3716:
3701:
3686:
3649:
3634:
3619:
3339:
3309:
3294:
3279:
2366:{\displaystyle \mathbf {a} =(a_{x},a_{y},a_{z}).}
2296:{\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3}).}
1997:), as illustrated in the figure. Here, the point
13343:
12834:; Leighton, R.; Sands, M. (2005). "Chapter 11".
12492:may be resolved into two components parallel to
12207:(2nd. ed.). London: Clarendon Press. 2001.
7140:with respect to a right-handed orthonormal basis
179:A vector is what is needed to "carry" the point
170:{\textstyle {\stackrel {\longrightarrow }{AB}}.}
12427:
12425:
1566:
828:) and same direction (e.g., the direction from
240:. These operations and associated laws qualify
12840:. Vol. I (2nd ed.). Addison Wesley.
11423:{\displaystyle {\mathbf {F} }=m{\mathbf {a} }}
5849:, discussed below, of the vector with itself:
3561:
2813:, respectively. In terms of these, any vector
863:is defined as a set to which is associated an
568:applied to a body has a point of contact (see
515:is defined more generally as any element of a
27:Geometric object that has length and direction
12915:
12543:
11378:of velocity. Its dimensions are length/time.
11365:of position. Its dimensions are length/time.
11282:. For constant velocity the position at time
11268:. Displacement has the dimensions of length.
6719:also becomes a right-handed system (although
5613:over vector addition in the following sense:
5042:This addition method is sometimes called the
5008:, and then drawing an arrow from the tail of
4985:The resulting vector is sometimes called the
4443:Opposite, parallel, and antiparallel vectors
3577:
1275:
425:included extensive treatment of the nabla or
12745:Kane, Thomas R.; Levinson, David A. (1996),
12744:
12577:
12450:
12422:
9833:" is the transpose of "rotation matrix from
9825:. This means that the "rotation matrix from
6668:unit vectors that are perpendicular to both
5016:. The new arrow drawn represents the vector
3855:
1627:(which is important to our understanding of
1235:
11594:. Unsourced material may be challenged and
11553:Learn how and when to remove these messages
11515:Vectors, pseudovectors, and transformations
7534:Conversion between multiple Cartesian bases
4801:
1674:. If you change units (a special case of a
12922:
12908:
12681:
12625:
12573:
12571:
12569:
12567:
12112:
11946:, which distinguishes between vectors and
10828:
8396:Distributing the dot-multiplication gives
544:A Euclidean vector may possess a definite
11687:Learn how and when to remove this message
11669:Learn how and when to remove this message
11653:, without removing the technical details.
11614:Learn how and when to remove this message
6614:
4771:
4429:
3813:to a surface (see figure). Moreover, the
3407:(note the difference in boldface). Thus,
1893:{\displaystyle {\underset {^{\sim }}{a}}}
1814:, or in lowercase italic boldface, as in
1550:
1198:
1107:
1059:
1027:
994:
645:
598:
453:, who was exposed to quaternions through
30:For mathematical vectors in general, see
12749:, Sunnyvale, California: OnLine Dynamics
12544:Harris, John W.; Stöcker, Horst (1998).
12480:coplanar with two non-collinear vectors
12415:, and its equivalence class is called a
11973:Covariance and contravariance of vectors
11883:. Ordinary vectors are sometimes called
11375:
11362:
7001:
6686:
6269:, or, since its result is a scalar, the
5910:
5705:|, which is not to be confused with the
5589:
5386:
5376:
3594:
2495:These numbers are often arranged into a
2070:
1916:(see figure), it can also be denoted as
1824:letters are typically used to represent
1684:covariance and contravariance of vectors
840:, which have no direction. For example,
760:. Such an equivalence class is called a
40:
12655:
12631:
12564:
12539:
12537:
12535:
12313:
12311:
12309:
12307:
12182:
8100:is to use column matrices along with a
7819:} that is not necessarily aligned with
5395:A vector may also be multiplied, or re-
5326:. This new arrow represents the vector
3776:
3761:
3698:
3683:
2055:On a two-dimensional diagram, a vector
1182:{\displaystyle {\overrightarrow {OP}}.}
804:{\displaystyle {\overrightarrow {AB}}.}
14:
13344:
13313:Comparison of linear algebra libraries
12803:
12725:
12705:Encyclopedic Dictionary of Mathematics
12589:
12101:
11381:
11016:The position vector has dimensions of
9851:the inverse is equal to the transpose;
5560:stretches a vector out by a factor of
5410:, these real numbers are often called
2213:{\displaystyle {\overrightarrow {OA}}}
1379:{\displaystyle {\overrightarrow {AB}}}
1316:. Other physical vectors, such as the
928:{\displaystyle {\overrightarrow {E}},}
890:{\displaystyle {\overrightarrow {E}},}
689:
12903:
12776:
12457:Thermodynamics and Differential Forms
12285:
12127:
11651:make it understandable to non-experts
10565:. In two dimensions this is simply a
9776:basis, the matrix containing all the
6392:that points in the same direction as
5556:Intuitively, multiplying by a scalar
2643:Another way to represent a vector in
1331:
967:{\displaystyle {\overrightarrow {E}}}
459:Treatise on Electricity and Magnetism
362:of equipollent directed segments. As
12809:
12754:
12532:
12387:
12304:
12257:
12255:
12253:
12251:
12226:
12224:
12177:participating institution membership
11625:
11592:adding citations to reliable sources
11559:
11518:
6757:The cross product can be written as
6691:An illustration of the cross product
6636:is the measure of the angle between
6540:, is a vector perpendicular to both
4451:if they have the same magnitude but
3573:
1354:in space determine the bound vector
859:. More precisely, a Euclidean space
12702:
12370:
12123:
11906:, and looking forward, each of the
10875:Position, velocity and acceleration
10804:
9872:
5082:. One can check geometrically that
2179:{\displaystyle \mathbf {a} =(2,3).}
1975:
867:of finite dimension over the reals
24:
12929:
11739:must be similarly transformed via
11374:of a point is vector which is the
9848:the determinant is unity, |C| = 1;
5360:
5026:
3348:{\displaystyle \mathbf {\hat {}} }
2623:
2082:
2047:
1726:
1716:
1649:
423:Elementary Treatise of Quaternions
251:Vectors play an important role in
25:
13398:
13377:Vectors (mathematics and physics)
12855:
12550:. Birkhäuser. Chapter 6, p. 332.
12248:
12232:"vector | Definition & Facts"
12221:
11898:One example of a pseudovector is
11534:This section has multiple issues.
7907:and the scalar components in the
7150:having the three vectors as rows
6998:instead of a vector (see below).
6528:) is only meaningful in three or
4820:of two vectors may be defined as
3862:Vector notation § Operations
2503:, particularly when dealing with
1256:is also directed rightward, then
1240:Since the physicist's concept of
1214:{\displaystyle \mathbb {R} ^{n},}
1123:{\displaystyle \mathbb {R} ^{n}.}
374:, Hamilton considered the vector
84:) is a geometric object that has
13326:
13325:
13303:Basic Linear Algebra Subprograms
13061:
12784:Geometry: A comprehensive course
12036:Tangential and normal components
11630:
11564:
11523:
11485:
11468:
11454:
11415:
11402:
11326:
11315:
11296:
11252:which specifies the position of
11228:
11182:
11136:
11096:
11086:
10992:
10965:
10938:
10917:
10758:
10746:
10678:
10651:
10618:
10591:
10527:
10481:
10431:
10385:
10332:
10305:
10278:
10251:
10214:
10187:
10160:
10133:
10082:
10036:
9987:
9960:
9927:
9900:
9700:
9685:
9652:
9637:
9604:
9589:
9556:
9541:
9508:
9493:
9460:
9445:
9412:
9397:
9364:
9349:
9316:
9301:
9266:
8731:
8716:
8698:
8683:
8665:
8650:
8618:
8603:
8585:
8570:
8552:
8537:
8505:
8490:
8472:
8457:
8439:
8424:
8367:
8349:
8331:
8313:
8278:
8260:
8242:
8224:
8189:
8171:
8153:
8135:
8018:
8009:
7981:
7972:
7944:
7935:
7885:
7867:
7849:
7837:
7765:
7756:
7728:
7719:
7691:
7682:
7629:
7611:
7593:
7581:
7571:is expressed, by definition, as
7511:
7503:
7495:
7478:
7470:
7462:
7445:
7437:
7429:
7412:
7404:
7396:
7382:
7374:
7366:
7352:
7344:
7336:
7177:
7169:
7161:
7093:
7085:
7074:
7063:
7055:
7047:
6968:
6902:
6836:
6776:
6766:
6616:
6591:
6578:
6566:
6558:
6505:
6417:
6409:
6324:
6311:
6299:
6291:
6141:
6129:
6099:
6087:
6057:
6045:
6015:
6006:
5991:
5885:
5877:
5863:
5736:
5531:
5497:
5463:
5435:
5277:
5233:
5189:
5151:
5143:
4962:
4918:
4874:
4836:
4828:
4650:
4623:
4596:
4575:
4541:
4514:
4487:
4466:
4321:
4294:
4267:
4246:
4216:
4189:
4162:
4141:
4101:
4074:
4047:
4026:
3968:
3924:
3880:
3837:to the radius and the latter is
3746:
3713:
3646:
3631:
3616:
3521:
3501:
3481:
3456:
3441:
3426:
3417:
3306:
3291:
3276:
3252:
3244:
3236:
3127:
3100:
3073:
3047:
3032:
3017:
3008:
2831:
2749:
2705:
2661:
2647:-dimensions is to introduce the
2515:
2393:
2311:
2241:
2151:
1800:
1777:
1754:
1072:{\displaystyle \mathbb {R} ^{n}}
1040:{\displaystyle \mathbb {R} ^{n}}
1007:{\displaystyle \mathbb {R} ^{n}}
738:being equipollent if the points
32:Vector (mathematics and physics)
13201:Seven-dimensional cross product
12837:The Feynman Lectures on Physics
12583:
12507:
12461:
12393:
12376:
12364:
12351:
12080:
11542:or discuss these issues on the
10908:whose base point is the origin
10785:seven-dimensional cross product
5365:The subtraction of two vectors
4741:
4711:
4402:
4375:
3866:The following section uses the
2122:, and are typically called the
1983:{\displaystyle {\mathfrak {a}}}
1742:Vectors are usually denoted in
1701:, a vector is any element of a
12760:Introduction to Linear Algebra
12279:
12197:
12132:
12117:
12106:
12095:
11735:, then a contravariant vector
11718:, so that a coordinate vector
11496:
11462:
11222:
11196:
11176:
11150:
11130:
11104:
10741:
10695:
10689:
10635:
10629:
10575:
10521:
10495:
10475:
10449:
10425:
10399:
10379:
10353:
10343:
10235:
10225:
10117:
10076:
10050:
10030:
10004:
9998:
9944:
9938:
9884:
8359:
8305:
8270:
8216:
8181:
8127:
7515:
7491:
7482:
7458:
7449:
7425:
7416:
7392:
7386:
7362:
7356:
7332:
7181:
7157:
7097:
7081:
7067:
7043:
6962:
6916:
6896:
6850:
6830:
6784:
6611:
6605:
6595:
6587:
6582:
6574:
6328:
6320:
6315:
6307:
6240:
6195:
6163:
6133:
6125:
6091:
6083:
6049:
6041:
6019:
6011:
5915:The normalization of a vector
5906:
5867:
5859:
5817:which is a consequence of the
5740:
5732:
5526:
5510:
5492:
5476:
5458:
5442:
5272:
5246:
5228:
5202:
5184:
5158:
4957:
4931:
4913:
4887:
4869:
4843:
4000:
3982:
3956:
3938:
3912:
3894:
2979:
2961:
2945:
2927:
2911:
2893:
2877:
2838:
2823:can be expressed in the form:
2781:
2763:
2737:
2719:
2693:
2675:
2619:
2579:
2477:
2400:
2357:
2318:
2287:
2248:
2170:
2158:
2143:= (2, 3) is simply written as
1935:
1841:
1709:and is often represented as a
1547:
1526:
1520:
1484:
1478:
1457:
1451:
1433:
646:
599:
499:. It is formally defined as a
157:
13:
1:
12620:
12205:The Oxford English Dictionary
11393:is the scalar multiplication
9726:By referring collectively to
7794:In another orthonormal basis
7654:The scalar components in the
6182:, and it is commonly denoted
5842:are orthogonal unit vectors.
3671:cylindrical coordinate system
1828:.) Other conventions include
1288:. For instance, the velocity
13043:Eigenvalues and eigenvectors
12399:In some old texts, the pair
12322:A History of Vector Analysis
5594:The scalar multiplications −
5066:is one of the diagonals. If
5031:The addition of two vectors
3833:of an object. The former is
1807:{\displaystyle \mathbf {w} }
1784:{\displaystyle \mathbf {v} }
1761:{\displaystyle \mathbf {u} }
1579:between two vectors. If the
1567:Euclidean and affine vectors
1284:, the magnitude of which is
1221:by mapping any point to the
852:are represented by vectors.
7:
12892:A conceptual introduction (
12868:Encyclopedia of Mathematics
12733:Encyclopedia of Mathematics
11936:
11432:Work is the dot product of
5408:conventional vector algebra
4127:
3868:Cartesian coordinate system
3734:spherical coordinate system
3562:Decomposition or resolution
2811:Cartesian coordinate system
2375:This can be generalised to
2120:Cartesian coordinate system
2100:Cartesian coordinate system
1731:Vector arrow pointing from
1338:Cartesian coordinate system
478:
463:Elements of Vector Analysis
263:of a moving object and the
10:
13403:
12762:(2nd ed.). Springer.
12682:Heinbockel, J. H. (2001),
12590:Rogers, Robert M. (2007).
10832:
10794:
10790:
10106:and in four dimensions as
7911:basis are, by definition,
7658:basis are, by definition,
7005:
6509:
6244:
6204:{\displaystyle {\vec {0}}}
6167:
5925:
5568:is an integer) as placing
5422:. The resulting vector is
5380:
4805:
3859:
3823:of a vector relate to the
3565:
1850:{\displaystyle {\vec {a}}}
1720:
1290:5 meters per second upward
1276:In physics and engineering
294:
29:
13321:
13283:
13239:
13176:
13128:
13070:
13059:
12955:
12937:
12164:Oxford English Dictionary
11895:of two ordinary vectors.
11701:coordinate transformation
5716:can be computed with the
5712:The length of the vector
5670:
5651:. One can also show that
5609:Scalar multiplication is
5004:at the head of the arrow
3856:Properties and operations
3225:(or scalar projections).
2114:). These numbers are the
1236:Examples in one dimension
415:Theorie der Ebbe und Flut
12880:Online vector identities
12578:Kane & Levinson 1996
12073:
11357:is the position at time
10879:The position of a point
10823:proportionality constant
10561:does, whose result is a
9226:). Each matrix element
5821:since the basis vectors
5024:, as illustrated below:
4802:Addition and subtraction
3850:inertial reference frame
3590:resolved with respect to
2079:with coordinates (2, 3).
1386:pointing from the point
1084:standard Euclidean space
756:, in this order, form a
437:William Kingdon Clifford
413:. Grassmann's 1840 work
12688:, Trafford Publishing,
12626:Mathematical treatments
12236:Encyclopedia Britannica
12169:Oxford University Press
12142:of vehere, "to carry"/
10829:Vector-valued functions
8102:direction cosine matrix
5338:being the opposite of
3550:is compatible with the
1638:(for free vectors) and
1304:, linear acceleration,
1142:, one may also find an
1098:to the Euclidean space
45:A vector pointing from
36:Vector (disambiguation)
34:. For other uses, see
13028:Row and column vectors
12726:Ivanov, A.B. (2001) ,
12437:Mathematics LibreTexts
12357:W. R. Hamilton (1846)
12067:Vector-valued function
11770:
11506:
11424:
11344:
11246:
11010:
10835:Vector-valued function
10775:
10549:
10100:
9718:
9182:
8963:
8752:
8388:
8039:
7899:
7786:
7646:
7525:
7317:
7107:
6986:
6692:
6624:
6497:
6374:trigonometric function
6360:is the measure of the
6348:
6263:(sometimes called the
6205:
6155:
5947:To normalize a vector
5923:
5898:
5809:
5701:‖ or, less commonly, |
5606:
5587:= 2) are given below:
5548:
5392:
5373:
5294:
5039:
4979:
4776:
4665:
4556:
4434:
4336:
4231:
4119:
4007:
3848:coordinate system, or
3789:
3726:
3659:
3600:
3568:Basis (linear algebra)
3533:
3349:
3319:
3260:
3145:
2992:
2791:
2635:
2487:
2367:
2297:
2214:
2180:
2139:= (0, 0) to the point
2087:
2080:
2052:
1984:
1946:
1894:
1851:
1808:
1785:
1762:
1739:
1621:pseudo-Euclidean space
1558:
1380:
1215:
1183:
1124:
1079:is often presented as
1073:
1041:
1008:
968:
929:
891:
805:
657:
610:
391:
354:) and a 3-dimensional
321:William Rowan Hamilton
171:
53:
13033:Row and column spaces
12978:Scalar multiplication
12292:mathworld.wolfram.com
11838:differential geometry
11771:
11769:{\displaystyle ^{-1}}
11507:
11425:
11361:= 0. Velocity is the
11345:
11247:
11011:
10776:
10550:
10101:
9787:transformation matrix
9719:
9183:
8964:
8753:
8389:
8040:
7900:
7787:
7647:
7526:
7318:
7108:
7014:scalar triple product
7008:Scalar triple product
7002:Scalar triple product
6987:
6690:
6625:
6498:
6349:
6206:
6156:
5914:
5899:
5810:
5593:
5549:
5420:scalar multiplication
5390:
5383:Scalar multiplication
5377:Scalar multiplication
5364:
5310:, place the tails of
5295:
5030:
4980:
4806:Further information:
4777:
4666:
4557:
4435:
4337:
4232:
4120:
4008:
3820:tangential components
3790:
3727:
3660:
3598:
3566:Further information:
3534:
3350:
3320:
3261:
3146:
2993:
2792:
2636:
2488:
2368:
2298:
2215:
2181:
2086:
2074:
2051:
1985:
1947:
1895:
1852:
1809:
1786:
1763:
1730:
1723:Vector representation
1721:Further information:
1559:
1381:
1216:
1184:
1148:Cartesian coordinates
1125:
1074:
1042:
1009:
984:real coordinate space
969:
930:
892:
806:
714:of points; two pairs
667:if the quadrilateral
658:
611:
501:directed line segment
407:Comte de Saint-Venant
386:
172:
121:directed line segment
44:
13168:Gram–Schmidt process
13120:Gaussian elimination
12703:Itô, Kiyosi (1993),
11842:contravariant vector
11751:
11705:contravariant vector
11588:improve this section
11443:
11397:
11290:
11081:
10912:
10572:
10110:
9881:
9275:
9210:) with those in the
8977:
8766:
8402:
8110:
7917:
7833:
7664:
7577:
7329:
7154:
7040:
7022:mixed triple product
6761:
6727:are not necessarily
6554:
6405:
6287:
6186:
5985:
5855:
5728:
5428:
5406:. In the context of
5342:, see drawing. And
5139:
5054:form the sides of a
4824:
4679:
4570:
4461:
4346:
4241:
4136:
4021:
3874:
3740:
3677:
3610:
3556:summation convention
3413:
3332:
3270:
3232:
3200:(see figure), while
3004:
2827:
2655:
2511:
2389:
2379:Euclidean space (or
2307:
2237:
2227:Euclidean space (or
2192:
2147:
1970:
1920:
1871:
1832:
1796:
1773:
1750:
1430:
1358:
1306:angular acceleration
1193:
1158:
1140:Gram–Schmidt process
1102:
1054:
1050:The Euclidean space
1022:
989:
951:
909:
871:
780:
620:
583:
467:Edwin Bidwell Wilson
451:Josiah Willard Gibbs
313:equivalence relation
206:algebraic operations
142:
112:units of measurement
76:(sometimes called a
13372:Concepts in physics
13298:Numerical stability
13178:Multilinear algebra
13153:Inner product space
13003:Linear independence
12894:applied mathematics
12890:Introducing Vectors
12804:Physical treatments
12339:on January 26, 2004
12286:Weisstein, Eric W.
12167:(Online ed.).
12021:Position (geometry)
11836:In the language of
11391:Newton's second law
11382:Force, energy, work
10843:. For instance, if
6703:is defined so that
6281:and is defined as:
5919:into a unit vector
5819:Pythagorean theorem
5799:
5781:
5763:
5709:(a scalar "norm").
4017:will be written as
3870:with basis vectors
3221:are the respective
1397:-axis to the point
865:inner product space
704:equivalence classes
690:Further information
455:James Clerk Maxwell
432:Elements of Dynamic
281:physical quantities
106:is a vector-valued
13387:Euclidean geometry
13008:Linear combination
12382:Formerly known as
12267:www.mathsisfun.com
12140:perfect participle
11766:
11722:is transformed to
11502:
11420:
11340:
11242:
11006:
10818:physical dimension
10771:
10545:
10543:
10096:
9714:
9712:
9178:
9169:
9136:
9007:
8959:
8957:
8748:
8746:
8384:
8382:
8035:
8033:
7895:
7782:
7780:
7642:
7521:
7313:
7307:
7135:are right-handed.
7121:linearly dependent
7103:
7034:) and defined as:
6982:
6695:The cross product
6693:
6660:which completes a
6620:
6548:and is defined as
6493:
6344:
6201:
6151:
5924:
5894:
5805:
5785:
5767:
5749:
5607:
5544:
5393:
5374:
5290:
5125:The difference of
5044:parallelogram rule
5040:
4975:
4772:
4661:
4552:
4453:opposite direction
4430:
4332:
4227:
4115:
4003:
3785:
3722:
3655:
3601:
3529:
3355:typically denotes
3345:
3315:
3256:
3182:vector projections
3141:
2988:
2787:
2631:
2570:
2483:
2363:
2293:
2210:
2176:
2130:scalar projections
2096:coordinate vectors
2088:
2081:
2053:
1980:
1942:
1890:
1888:
1847:
1804:
1781:
1758:
1740:
1629:special relativity
1554:
1376:
1332:In Cartesian space
1252:. If the positive
1211:
1179:
1120:
1069:
1037:
1014:equipped with the
1004:
964:
925:
887:
801:
700:synthetic geometry
696:Euclidean geometry
653:
606:
497:relative direction
419:Peter Guthrie Tait
370:to complement the
319:was introduced by
167:
118:, formulated as a
54:
13382:Analytic geometry
13339:
13338:
13206:Geometric algebra
13163:Kronecker product
12998:Linear projection
12983:Vector projection
12847:978-0-8053-9046-9
12823:978-0-486-66110-0
12810:Aris, R. (1990).
12718:978-0-262-59020-4
12674:978-0-471-00007-5
12648:978-0-471-00005-1
12175:(Subscription or
11968:Coordinate system
11713:invertible matrix
11697:
11696:
11689:
11679:
11678:
11671:
11624:
11623:
11616:
11557:
11023:Given two points
9785:is known as the "
7509:
7501:
7476:
7468:
7443:
7435:
7410:
7402:
7380:
7372:
7350:
7342:
7175:
7167:
7061:
7053:
7016:(also called the
6648:is a unit vector
6520:(also called the
6198:
6137:
6095:
6053:
6023:
5997:
5889:
5800:
4455:; so two vectors
3964:
3920:
3782:
3767:
3752:
3719:
3704:
3689:
3652:
3637:
3622:
3342:
3338:
3312:
3297:
3282:
3223:scalar components
3177:vector components
2987:
2745:
2701:
2607:
2594:
2225:three dimensional
2208:
2125:scalar components
1939:
1875:
1844:
1711:coordinate vector
1374:
1230:coordinate vector
1174:
1144:orthonormal basis
962:
920:
882:
796:
776:is often denoted
650:
603:
503:, or arrow, in a
435:was published by
399:Hermann Grassmann
384:of a quaternion:
327:, which is a sum
301:Giusto Bellavitis
277:coordinate system
187:; the Latin word
161:
138:, and denoted by
108:physical quantity
16:(Redirected from
13394:
13357:Abstract algebra
13329:
13328:
13211:Exterior algebra
13148:Hadamard product
13065:
13053:Linear equations
12924:
12917:
12910:
12901:
12900:
12876:
12851:
12832:Feynman, Richard
12827:
12799:
12787:
12773:
12750:
12740:
12721:
12707:(2nd ed.),
12698:
12678:
12666:
12652:
12614:
12613:
12587:
12581:
12580:, pp. 20–22
12575:
12562:
12561:
12541:
12530:
12529:
12527:
12526:
12517:. Archived from
12511:
12505:
12503:
12497:
12491:
12485:
12479:
12465:
12459:
12454:
12448:
12447:
12445:
12444:
12429:
12420:
12410:
12397:
12391:
12380:
12374:
12368:
12362:
12361:3rd series 29 27
12355:
12349:
12347:
12345:
12344:
12338:
12332:. Archived from
12331:
12318:Michael J. Crowe
12315:
12302:
12301:
12299:
12298:
12283:
12277:
12276:
12274:
12273:
12259:
12246:
12245:
12243:
12242:
12228:
12219:
12218:
12201:
12195:
12194:
12192:
12191:
12180:
12172:
12160:
12136:
12130:
12126:, p. 1678;
12121:
12115:
12110:
12104:
12099:
12087:
12084:
12016:Parity (physics)
11996:Ausdehnungslehre
11958:Clifford algebra
11900:angular velocity
11784:consists of the
11779:
11775:
11773:
11772:
11767:
11765:
11764:
11734:
11692:
11685:
11674:
11667:
11663:
11660:
11654:
11634:
11633:
11626:
11619:
11612:
11608:
11605:
11599:
11568:
11560:
11549:
11527:
11526:
11519:
11511:
11509:
11508:
11503:
11495:
11494:
11489:
11488:
11478:
11477:
11472:
11471:
11458:
11457:
11429:
11427:
11426:
11421:
11419:
11418:
11406:
11405:
11349:
11347:
11346:
11341:
11336:
11335:
11330:
11329:
11319:
11318:
11306:
11305:
11300:
11299:
11251:
11249:
11248:
11243:
11238:
11237:
11232:
11231:
11221:
11220:
11208:
11207:
11192:
11191:
11186:
11185:
11175:
11174:
11162:
11161:
11146:
11145:
11140:
11139:
11129:
11128:
11116:
11115:
11100:
11099:
11090:
11089:
11015:
11013:
11012:
11007:
11002:
11001:
10996:
10995:
10988:
10987:
10975:
10974:
10969:
10968:
10961:
10960:
10948:
10947:
10942:
10941:
10934:
10933:
10921:
10920:
10805:Length and units
10780:
10778:
10777:
10772:
10767:
10766:
10761:
10755:
10754:
10749:
10740:
10739:
10730:
10729:
10717:
10716:
10707:
10706:
10688:
10687:
10682:
10681:
10674:
10673:
10661:
10660:
10655:
10654:
10647:
10646:
10628:
10627:
10622:
10621:
10614:
10613:
10601:
10600:
10595:
10594:
10587:
10586:
10559:exterior product
10554:
10552:
10551:
10546:
10544:
10537:
10536:
10531:
10530:
10520:
10519:
10507:
10506:
10491:
10490:
10485:
10484:
10474:
10473:
10461:
10460:
10441:
10440:
10435:
10434:
10424:
10423:
10411:
10410:
10395:
10394:
10389:
10388:
10378:
10377:
10365:
10364:
10342:
10341:
10336:
10335:
10328:
10327:
10315:
10314:
10309:
10308:
10301:
10300:
10288:
10287:
10282:
10281:
10274:
10273:
10261:
10260:
10255:
10254:
10247:
10246:
10224:
10223:
10218:
10217:
10210:
10209:
10197:
10196:
10191:
10190:
10183:
10182:
10170:
10169:
10164:
10163:
10156:
10155:
10143:
10142:
10137:
10136:
10129:
10128:
10105:
10103:
10102:
10097:
10092:
10091:
10086:
10085:
10075:
10074:
10062:
10061:
10046:
10045:
10040:
10039:
10029:
10028:
10016:
10015:
9997:
9996:
9991:
9990:
9983:
9982:
9970:
9969:
9964:
9963:
9956:
9955:
9937:
9936:
9931:
9930:
9923:
9922:
9910:
9909:
9904:
9903:
9896:
9895:
9873:Other dimensions
9821:is equal to its
9723:
9721:
9720:
9715:
9713:
9709:
9708:
9703:
9694:
9693:
9688:
9675:
9674:
9661:
9660:
9655:
9646:
9645:
9640:
9627:
9626:
9613:
9612:
9607:
9598:
9597:
9592:
9579:
9578:
9565:
9564:
9559:
9550:
9549:
9544:
9531:
9530:
9517:
9516:
9511:
9502:
9501:
9496:
9483:
9482:
9469:
9468:
9463:
9454:
9453:
9448:
9435:
9434:
9421:
9420:
9415:
9406:
9405:
9400:
9387:
9386:
9373:
9372:
9367:
9358:
9357:
9352:
9339:
9338:
9325:
9324:
9319:
9310:
9309:
9304:
9291:
9290:
9259:direction cosine
9237:direction cosine
9187:
9185:
9184:
9179:
9174:
9173:
9141:
9140:
9133:
9132:
9121:
9120:
9109:
9108:
9095:
9094:
9083:
9082:
9071:
9070:
9057:
9056:
9045:
9044:
9033:
9032:
9012:
9011:
8968:
8966:
8965:
8960:
8958:
8948:
8947:
8932:
8931:
8916:
8915:
8886:
8885:
8870:
8869:
8854:
8853:
8824:
8823:
8808:
8807:
8792:
8791:
8757:
8755:
8754:
8749:
8747:
8740:
8739:
8734:
8725:
8724:
8719:
8707:
8706:
8701:
8692:
8691:
8686:
8674:
8673:
8668:
8659:
8658:
8653:
8627:
8626:
8621:
8612:
8611:
8606:
8594:
8593:
8588:
8579:
8578:
8573:
8561:
8560:
8555:
8546:
8545:
8540:
8514:
8513:
8508:
8499:
8498:
8493:
8481:
8480:
8475:
8466:
8465:
8460:
8448:
8447:
8442:
8433:
8432:
8427:
8393:
8391:
8390:
8385:
8383:
8376:
8375:
8370:
8358:
8357:
8352:
8340:
8339:
8334:
8322:
8321:
8316:
8287:
8286:
8281:
8269:
8268:
8263:
8251:
8250:
8245:
8233:
8232:
8227:
8198:
8197:
8192:
8180:
8179:
8174:
8162:
8161:
8156:
8144:
8143:
8138:
8044:
8042:
8041:
8036:
8034:
8027:
8026:
8021:
8012:
7990:
7989:
7984:
7975:
7953:
7952:
7947:
7938:
7904:
7902:
7901:
7896:
7894:
7893:
7888:
7876:
7875:
7870:
7858:
7857:
7852:
7840:
7827:is expressed as
7791:
7789:
7788:
7783:
7781:
7774:
7773:
7768:
7759:
7737:
7736:
7731:
7722:
7700:
7699:
7694:
7685:
7651:
7649:
7648:
7643:
7638:
7637:
7632:
7620:
7619:
7614:
7602:
7601:
7596:
7584:
7567:basis, a vector
7530:
7528:
7527:
7522:
7514:
7507:
7506:
7499:
7498:
7481:
7474:
7473:
7466:
7465:
7448:
7441:
7440:
7433:
7432:
7415:
7408:
7407:
7400:
7399:
7385:
7378:
7377:
7370:
7369:
7355:
7348:
7347:
7340:
7339:
7322:
7320:
7319:
7314:
7312:
7311:
7304:
7303:
7292:
7291:
7280:
7279:
7266:
7265:
7254:
7253:
7242:
7241:
7228:
7227:
7216:
7215:
7204:
7203:
7180:
7173:
7172:
7165:
7164:
7112:
7110:
7109:
7104:
7096:
7088:
7077:
7066:
7059:
7058:
7051:
7050:
6991:
6989:
6988:
6983:
6978:
6977:
6972:
6971:
6961:
6960:
6951:
6950:
6938:
6937:
6928:
6927:
6912:
6911:
6906:
6905:
6895:
6894:
6885:
6884:
6872:
6871:
6862:
6861:
6846:
6845:
6840:
6839:
6829:
6828:
6819:
6818:
6806:
6805:
6796:
6795:
6780:
6779:
6770:
6769:
6629:
6627:
6626:
6621:
6619:
6598:
6594:
6585:
6581:
6569:
6561:
6502:
6500:
6499:
6494:
6489:
6488:
6479:
6478:
6466:
6465:
6456:
6455:
6443:
6442:
6433:
6432:
6420:
6412:
6353:
6351:
6350:
6345:
6331:
6327:
6318:
6314:
6302:
6294:
6273:) is denoted by
6236:
6210:
6208:
6207:
6202:
6200:
6199:
6191:
6181:
6160:
6158:
6157:
6152:
6150:
6149:
6144:
6138:
6136:
6132:
6123:
6122:
6113:
6108:
6107:
6102:
6096:
6094:
6090:
6081:
6080:
6071:
6066:
6065:
6060:
6054:
6052:
6048:
6039:
6038:
6029:
6024:
6022:
6018:
6009:
6004:
5999:
5998:
5990:
5974:
5903:
5901:
5900:
5895:
5890:
5888:
5880:
5875:
5870:
5866:
5814:
5812:
5811:
5806:
5801:
5798:
5793:
5780:
5775:
5762:
5757:
5748:
5743:
5739:
5647:and all scalars
5639:for all vectors
5553:
5551:
5550:
5545:
5540:
5539:
5534:
5525:
5524:
5506:
5505:
5500:
5491:
5490:
5472:
5471:
5466:
5457:
5456:
5438:
5299:
5297:
5296:
5291:
5286:
5285:
5280:
5271:
5270:
5258:
5257:
5242:
5241:
5236:
5227:
5226:
5214:
5213:
5198:
5197:
5192:
5183:
5182:
5170:
5169:
5154:
5146:
4987:resultant vector
4984:
4982:
4981:
4976:
4971:
4970:
4965:
4956:
4955:
4943:
4942:
4927:
4926:
4921:
4912:
4911:
4899:
4898:
4883:
4882:
4877:
4868:
4867:
4855:
4854:
4839:
4831:
4784:Two vectors are
4781:
4779:
4778:
4773:
4767:
4766:
4751:
4750:
4737:
4736:
4721:
4720:
4707:
4706:
4691:
4690:
4673:are opposite if
4670:
4668:
4667:
4662:
4660:
4659:
4654:
4653:
4646:
4645:
4633:
4632:
4627:
4626:
4619:
4618:
4606:
4605:
4600:
4599:
4592:
4591:
4579:
4578:
4561:
4559:
4558:
4553:
4551:
4550:
4545:
4544:
4537:
4536:
4524:
4523:
4518:
4517:
4510:
4509:
4497:
4496:
4491:
4490:
4483:
4482:
4470:
4469:
4447:Two vectors are
4439:
4437:
4436:
4431:
4425:
4424:
4412:
4411:
4398:
4397:
4385:
4384:
4371:
4370:
4358:
4357:
4341:
4339:
4338:
4333:
4331:
4330:
4325:
4324:
4317:
4316:
4304:
4303:
4298:
4297:
4290:
4289:
4277:
4276:
4271:
4270:
4263:
4262:
4250:
4249:
4236:
4234:
4233:
4228:
4226:
4225:
4220:
4219:
4212:
4211:
4199:
4198:
4193:
4192:
4185:
4184:
4172:
4171:
4166:
4165:
4158:
4157:
4145:
4144:
4124:
4122:
4121:
4116:
4111:
4110:
4105:
4104:
4097:
4096:
4084:
4083:
4078:
4077:
4070:
4069:
4057:
4056:
4051:
4050:
4043:
4042:
4030:
4029:
4012:
4010:
4009:
4004:
3978:
3977:
3972:
3971:
3962:
3934:
3933:
3928:
3927:
3918:
3890:
3889:
3884:
3883:
3794:
3792:
3791:
3786:
3784:
3783:
3775:
3769:
3768:
3760:
3754:
3753:
3745:
3731:
3729:
3728:
3723:
3721:
3720:
3712:
3706:
3705:
3697:
3691:
3690:
3682:
3664:
3662:
3661:
3656:
3654:
3653:
3645:
3639:
3638:
3630:
3624:
3623:
3615:
3538:
3536:
3535:
3530:
3525:
3524:
3518:
3517:
3505:
3504:
3498:
3497:
3485:
3484:
3478:
3477:
3465:
3464:
3459:
3450:
3449:
3444:
3435:
3434:
3429:
3420:
3354:
3352:
3351:
3346:
3344:
3343:
3337:
3324:
3322:
3321:
3316:
3314:
3313:
3305:
3299:
3298:
3290:
3284:
3283:
3275:
3265:
3263:
3262:
3257:
3255:
3247:
3239:
3150:
3148:
3147:
3142:
3137:
3136:
3131:
3130:
3123:
3122:
3110:
3109:
3104:
3103:
3096:
3095:
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3082:
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3076:
3069:
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3050:
3041:
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3011:
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2989:
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2617:
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2592:
2591:
2590:
2575:
2574:
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2566:
2553:
2552:
2539:
2538:
2518:
2492:
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2484:
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2475:
2463:
2462:
2438:
2437:
2425:
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2412:
2411:
2396:
2384:
2372:
2370:
2369:
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2356:
2355:
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2342:
2330:
2329:
2314:
2302:
2300:
2299:
2294:
2286:
2285:
2273:
2272:
2260:
2259:
2244:
2232:
2219:
2217:
2216:
2211:
2209:
2204:
2196:
2185:
2183:
2182:
2177:
2154:
2017:, and the point
1989:
1987:
1986:
1981:
1979:
1978:
1966:letters such as
1951:
1949:
1948:
1943:
1941:
1940:
1938:
1933:
1925:
1899:
1897:
1896:
1891:
1889:
1887:
1886:
1856:
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1853:
1848:
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1845:
1837:
1813:
1811:
1810:
1805:
1803:
1790:
1788:
1787:
1782:
1780:
1767:
1765:
1764:
1759:
1757:
1746:boldface, as in
1602:exterior product
1596:in space of the
1563:
1561:
1560:
1555:
1417:
1403:
1392:
1385:
1383:
1382:
1377:
1375:
1370:
1362:
1353:
1346:
1314:angular momentum
1225:
1220:
1218:
1217:
1212:
1207:
1206:
1201:
1188:
1186:
1185:
1180:
1175:
1170:
1162:
1153:
1133:
1129:
1127:
1126:
1121:
1116:
1115:
1110:
1093:
1089:
1078:
1076:
1075:
1070:
1068:
1067:
1062:
1046:
1044:
1043:
1038:
1036:
1035:
1030:
1013:
1011:
1010:
1005:
1003:
1002:
997:
973:
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970:
965:
963:
955:
934:
932:
931:
926:
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913:
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894:
893:
888:
883:
875:
862:
835:
831:
827:
810:
808:
807:
802:
797:
792:
784:
775:
755:
737:
725:
662:
660:
659:
654:
652:
651:
649:
643:
642:
634:
625:
615:
613:
612:
607:
605:
604:
602:
596:
588:
509:pure mathematics
379:
349:
340:
176:
174:
173:
168:
163:
162:
160:
155:
147:
78:geometric vector
70:Euclidean vector
21:
13402:
13401:
13397:
13396:
13395:
13393:
13392:
13391:
13362:Vector calculus
13342:
13341:
13340:
13335:
13317:
13279:
13235:
13172:
13124:
13066:
13057:
13023:Change of basis
13013:Multilinear map
12951:
12933:
12928:
12861:
12858:
12848:
12824:
12806:
12796:
12770:
12747:Dynamics Online
12719:
12696:
12675:
12649:
12628:
12623:
12618:
12617:
12602:
12588:
12584:
12576:
12565:
12558:
12542:
12533:
12524:
12522:
12513:
12512:
12508:
12499:
12493:
12487:
12481:
12475:
12466:
12462:
12455:
12451:
12442:
12440:
12431:
12430:
12423:
12400:
12398:
12394:
12381:
12377:
12369:
12365:
12356:
12352:
12348:on the subject.
12342:
12340:
12336:
12329:
12327:"lecture notes"
12325:
12324:; see also his
12316:
12305:
12296:
12294:
12284:
12280:
12271:
12269:
12261:
12260:
12249:
12240:
12238:
12230:
12229:
12222:
12215:
12203:
12202:
12198:
12189:
12187:
12174:
12138:Latin: vectus,
12137:
12133:
12122:
12118:
12113:Heinbockel 2001
12111:
12107:
12100:
12096:
12091:
12090:
12085:
12081:
12076:
12071:
12062:Vector notation
12057:Vector calculus
11939:
11902:. Driving in a
11796:-components of
11757:
11754:
11752:
11749:
11748:
11740:
11723:
11693:
11682:
11681:
11680:
11675:
11664:
11658:
11655:
11647:help improve it
11644:
11635:
11631:
11620:
11609:
11603:
11600:
11585:
11569:
11528:
11524:
11517:
11490:
11484:
11483:
11482:
11473:
11467:
11466:
11465:
11453:
11452:
11444:
11441:
11440:
11414:
11413:
11401:
11400:
11398:
11395:
11394:
11384:
11376:time derivative
11363:time derivative
11356:
11331:
11325:
11324:
11323:
11314:
11313:
11301:
11295:
11294:
11293:
11291:
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11287:
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11226:
11225:
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11212:
11203:
11199:
11187:
11181:
11180:
11179:
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11166:
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11141:
11135:
11134:
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11120:
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11107:
11095:
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11084:
11082:
11079:
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11072:
11065:
11058:
11047:
11040:
11033:
10997:
10991:
10990:
10989:
10983:
10979:
10970:
10964:
10963:
10962:
10956:
10952:
10943:
10937:
10936:
10935:
10929:
10925:
10916:
10915:
10913:
10910:
10909:
10906:position vector
10903:
10896:
10889:
10877:
10837:
10831:
10807:
10799:
10797:Vector quantity
10793:
10762:
10757:
10756:
10750:
10745:
10744:
10735:
10731:
10725:
10721:
10712:
10708:
10702:
10698:
10683:
10677:
10676:
10675:
10669:
10665:
10656:
10650:
10649:
10648:
10642:
10638:
10623:
10617:
10616:
10615:
10609:
10605:
10596:
10590:
10589:
10588:
10582:
10578:
10573:
10570:
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10469:
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10360:
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10331:
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10319:
10310:
10304:
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10296:
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10228:
10219:
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10192:
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9951:
9947:
9932:
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9918:
9914:
9905:
9899:
9898:
9897:
9891:
9887:
9882:
9879:
9878:
9875:
9799:rotation matrix
9784:
9771:
9764:
9757:
9746:
9739:
9732:
9711:
9710:
9704:
9699:
9698:
9689:
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9670:
9666:
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9470:
9464:
9459:
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9449:
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9426:
9423:
9422:
9416:
9411:
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9401:
9396:
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9378:
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9128:
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9112:
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9104:
9100:
9097:
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9086:
9084:
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9040:
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9028:
9024:
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8991:
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8956:
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8927:
8923:
8911:
8907:
8900:
8894:
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8865:
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8845:
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8764:
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8745:
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8714:
8702:
8697:
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8687:
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8617:
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8574:
8569:
8568:
8556:
8551:
8550:
8541:
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8519:
8518:
8509:
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8494:
8489:
8488:
8476:
8471:
8470:
8461:
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8443:
8438:
8437:
8428:
8423:
8422:
8412:
8405:
8403:
8400:
8399:
8381:
8380:
8371:
8366:
8365:
8353:
8348:
8347:
8335:
8330:
8329:
8317:
8312:
8311:
8298:
8292:
8291:
8282:
8277:
8276:
8264:
8259:
8258:
8246:
8241:
8240:
8228:
8223:
8222:
8209:
8203:
8202:
8193:
8188:
8187:
8175:
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8169:
8157:
8152:
8151:
8139:
8134:
8133:
8120:
8113:
8111:
8108:
8107:
8032:
8031:
8022:
8017:
8016:
8008:
8001:
7995:
7994:
7985:
7980:
7979:
7971:
7964:
7958:
7957:
7948:
7943:
7942:
7934:
7927:
7920:
7918:
7915:
7914:
7889:
7884:
7883:
7871:
7866:
7865:
7853:
7848:
7847:
7836:
7834:
7831:
7830:
7818:
7811:
7804:
7779:
7778:
7769:
7764:
7763:
7755:
7748:
7742:
7741:
7732:
7727:
7726:
7718:
7711:
7705:
7704:
7695:
7690:
7689:
7681:
7674:
7667:
7665:
7662:
7661:
7633:
7628:
7627:
7615:
7610:
7609:
7597:
7592:
7591:
7580:
7578:
7575:
7574:
7562:
7555:
7548:
7536:
7510:
7502:
7494:
7477:
7469:
7461:
7444:
7436:
7428:
7411:
7403:
7395:
7381:
7373:
7365:
7351:
7343:
7335:
7330:
7327:
7326:
7306:
7305:
7299:
7295:
7293:
7287:
7283:
7281:
7275:
7271:
7268:
7267:
7261:
7257:
7255:
7249:
7245:
7243:
7237:
7233:
7230:
7229:
7223:
7219:
7217:
7211:
7207:
7205:
7199:
7195:
7188:
7187:
7176:
7168:
7160:
7155:
7152:
7151:
7138:In components (
7092:
7084:
7073:
7062:
7054:
7046:
7041:
7038:
7037:
7010:
7004:
6973:
6967:
6966:
6965:
6956:
6952:
6946:
6942:
6933:
6929:
6923:
6919:
6907:
6901:
6900:
6899:
6890:
6886:
6880:
6876:
6867:
6863:
6857:
6853:
6841:
6835:
6834:
6833:
6824:
6820:
6814:
6810:
6801:
6797:
6791:
6787:
6775:
6774:
6765:
6764:
6762:
6759:
6758:
6733:right-hand rule
6731:). This is the
6615:
6590:
6586:
6577:
6573:
6565:
6557:
6555:
6552:
6551:
6514:
6508:
6484:
6480:
6474:
6470:
6461:
6457:
6451:
6447:
6438:
6434:
6428:
6424:
6416:
6408:
6406:
6403:
6402:
6323:
6319:
6310:
6306:
6298:
6290:
6288:
6285:
6284:
6255:of two vectors
6249:
6243:
6224:
6190:
6189:
6187:
6184:
6183:
6179:
6172:
6166:
6145:
6140:
6139:
6128:
6124:
6118:
6114:
6112:
6103:
6098:
6097:
6086:
6082:
6076:
6072:
6070:
6061:
6056:
6055:
6044:
6040:
6034:
6030:
6028:
6014:
6010:
6005:
6003:
5989:
5988:
5986:
5983:
5982:
5972:
5965:
5958:
5948:
5930:
5909:
5884:
5876:
5874:
5862:
5858:
5856:
5853:
5852:
5841:
5834:
5827:
5794:
5789:
5776:
5771:
5758:
5753:
5747:
5735:
5731:
5729:
5726:
5725:
5697:is denoted by ‖
5673:
5535:
5530:
5529:
5520:
5516:
5501:
5496:
5495:
5486:
5482:
5467:
5462:
5461:
5452:
5448:
5434:
5429:
5426:
5425:
5385:
5379:
5322:to the head of
5281:
5276:
5275:
5266:
5262:
5253:
5249:
5237:
5232:
5231:
5222:
5218:
5209:
5205:
5193:
5188:
5187:
5178:
5174:
5165:
5161:
5150:
5142:
5140:
5137:
5136:
5012:to the head of
4966:
4961:
4960:
4951:
4947:
4938:
4934:
4922:
4917:
4916:
4907:
4903:
4894:
4890:
4878:
4873:
4872:
4863:
4859:
4850:
4846:
4835:
4827:
4825:
4822:
4821:
4810:
4804:
4787:equidirectional
4762:
4758:
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4742:
4732:
4728:
4716:
4712:
4702:
4698:
4686:
4682:
4680:
4677:
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4641:
4637:
4628:
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4528:
4519:
4513:
4512:
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4505:
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4484:
4478:
4474:
4465:
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4462:
4459:
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4445:
4420:
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4393:
4389:
4380:
4376:
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4326:
4320:
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4264:
4258:
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4221:
4215:
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4213:
4207:
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4099:
4098:
4092:
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3965:
3929:
3923:
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3885:
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3872:
3871:
3864:
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3711:
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3696:
3695:
3681:
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3629:
3628:
3614:
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3608:
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3549:
3520:
3519:
3513:
3509:
3500:
3499:
3493:
3489:
3480:
3479:
3473:
3469:
3460:
3455:
3454:
3445:
3440:
3439:
3430:
3425:
3424:
3416:
3414:
3411:
3410:
3406:
3397:
3388:
3378:
3371:
3364:
3336:
3335:
3333:
3330:
3329:
3325:, in which the
3304:
3303:
3289:
3288:
3274:
3273:
3271:
3268:
3267:
3251:
3243:
3235:
3233:
3230:
3229:
3220:
3213:
3206:
3174:are called the
3173:
3166:
3159:
3132:
3126:
3125:
3124:
3118:
3114:
3105:
3099:
3098:
3097:
3091:
3087:
3078:
3072:
3071:
3070:
3064:
3060:
3051:
3046:
3045:
3036:
3031:
3030:
3021:
3016:
3015:
3007:
3005:
3002:
3001:
2955:
2951:
2921:
2917:
2887:
2883:
2871:
2867:
2858:
2854:
2845:
2841:
2830:
2828:
2825:
2824:
2818:
2754:
2748:
2747:
2746:
2710:
2704:
2703:
2702:
2666:
2660:
2659:
2658:
2656:
2653:
2652:
2622:
2618:
2612:
2608:
2599:
2595:
2586:
2582:
2569:
2568:
2562:
2558:
2555:
2554:
2548:
2544:
2541:
2540:
2534:
2530:
2523:
2522:
2514:
2512:
2509:
2508:
2471:
2467:
2452:
2448:
2433:
2429:
2420:
2416:
2407:
2403:
2392:
2390:
2387:
2386:
2380:
2351:
2347:
2338:
2334:
2325:
2321:
2310:
2308:
2305:
2304:
2281:
2277:
2268:
2264:
2255:
2251:
2240:
2238:
2235:
2234:
2228:
2197:
2195:
2193:
2190:
2189:
2150:
2148:
2145:
2144:
1974:
1973:
1971:
1968:
1967:
1934:
1926:
1924:
1923:
1921:
1918:
1917:
1882:
1879:
1874:
1872:
1869:
1868:
1836:
1835:
1833:
1830:
1829:
1799:
1797:
1794:
1793:
1776:
1774:
1771:
1770:
1753:
1751:
1748:
1747:
1725:
1719:
1717:Representations
1676:change of basis
1652:
1650:Generalizations
1625:Minkowski space
1569:
1431:
1428:
1427:
1412:
1398:
1387:
1363:
1361:
1359:
1356:
1355:
1348:
1341:
1334:
1310:linear momentum
1278:
1238:
1223:
1202:
1197:
1196:
1194:
1191:
1190:
1163:
1161:
1159:
1156:
1155:
1151:
1131:
1111:
1106:
1105:
1103:
1100:
1099:
1091:
1087:
1063:
1058:
1057:
1055:
1052:
1051:
1031:
1026:
1025:
1023:
1020:
1019:
998:
993:
992:
990:
987:
986:
954:
952:
949:
948:
912:
910:
907:
906:
874:
872:
869:
868:
860:
833:
829:
817:
785:
783:
781:
778:
777:
765:
739:
727:
715:
692:
644:
635:
627:
626:
624:
623:
621:
618:
617:
597:
589:
587:
586:
584:
581:
580:
570:resultant force
521:Euclidean space
505:Euclidean space
481:
472:Vector Analysis
411:Matthew O'Brien
395:Augustin Cauchy
375:
364:complex numbers
345:
328:
297:
156:
148:
146:
145:
143:
140:
139:
114:and possibly a
103:vector quantity
39:
28:
23:
22:
18:Vector addition
15:
12:
11:
5:
13400:
13390:
13389:
13384:
13379:
13374:
13369:
13367:Linear algebra
13364:
13359:
13354:
13337:
13336:
13334:
13333:
13322:
13319:
13318:
13316:
13315:
13310:
13305:
13300:
13295:
13293:Floating-point
13289:
13287:
13281:
13280:
13278:
13277:
13275:Tensor product
13272:
13267:
13262:
13260:Function space
13257:
13252:
13246:
13244:
13237:
13236:
13234:
13233:
13228:
13223:
13218:
13213:
13208:
13203:
13198:
13196:Triple product
13193:
13188:
13182:
13180:
13174:
13173:
13171:
13170:
13165:
13160:
13155:
13150:
13145:
13140:
13134:
13132:
13126:
13125:
13123:
13122:
13117:
13112:
13110:Transformation
13107:
13102:
13100:Multiplication
13097:
13092:
13087:
13082:
13076:
13074:
13068:
13067:
13060:
13058:
13056:
13055:
13050:
13045:
13040:
13035:
13030:
13025:
13020:
13015:
13010:
13005:
13000:
12995:
12990:
12985:
12980:
12975:
12970:
12965:
12959:
12957:
12956:Basic concepts
12953:
12952:
12950:
12949:
12944:
12938:
12935:
12934:
12931:Linear algebra
12927:
12926:
12919:
12912:
12904:
12898:
12897:
12887:
12877:
12857:
12856:External links
12854:
12853:
12852:
12846:
12828:
12822:
12805:
12802:
12801:
12800:
12794:
12774:
12768:
12752:
12742:
12723:
12717:
12700:
12694:
12679:
12673:
12653:
12647:
12627:
12624:
12622:
12619:
12616:
12615:
12600:
12582:
12563:
12556:
12531:
12506:
12460:
12449:
12433:"1.1: Vectors"
12421:
12392:
12384:located vector
12375:
12373:, p. 1678
12363:
12350:
12303:
12278:
12247:
12220:
12213:
12196:
12131:
12116:
12105:
12093:
12092:
12089:
12088:
12078:
12077:
12075:
12072:
12070:
12069:
12064:
12059:
12054:
12049:
12044:
12039:
12033:
12028:
12023:
12018:
12013:
12008:
12003:
11998:
11989:
11987:Function space
11984:
11975:
11970:
11965:
11963:Complex number
11960:
11955:
11950:
11940:
11938:
11935:
11920:magnetic field
11917:
11866:
11854:tangent vector
11819:electric field
11763:
11760:
11756:
11695:
11694:
11677:
11676:
11638:
11636:
11629:
11622:
11621:
11572:
11570:
11563:
11558:
11532:
11531:
11529:
11522:
11516:
11513:
11501:
11498:
11493:
11487:
11481:
11476:
11470:
11464:
11461:
11456:
11451:
11448:
11417:
11412:
11409:
11404:
11383:
11380:
11354:
11339:
11334:
11328:
11322:
11317:
11312:
11309:
11304:
11298:
11241:
11236:
11230:
11224:
11219:
11215:
11211:
11206:
11202:
11198:
11195:
11190:
11184:
11178:
11173:
11169:
11165:
11160:
11156:
11152:
11149:
11144:
11138:
11132:
11127:
11123:
11119:
11114:
11110:
11106:
11103:
11098:
11093:
11088:
11070:
11063:
11056:
11045:
11038:
11031:
11005:
11000:
10994:
10986:
10982:
10978:
10973:
10967:
10959:
10955:
10951:
10946:
10940:
10932:
10928:
10924:
10919:
10901:
10894:
10887:
10876:
10873:
10861:differentiated
10833:Main article:
10830:
10827:
10806:
10803:
10795:Main article:
10792:
10789:
10770:
10765:
10760:
10753:
10748:
10743:
10738:
10734:
10728:
10724:
10720:
10715:
10711:
10705:
10701:
10697:
10694:
10691:
10686:
10680:
10672:
10668:
10664:
10659:
10653:
10645:
10641:
10637:
10634:
10631:
10626:
10620:
10612:
10608:
10604:
10599:
10593:
10585:
10581:
10577:
10540:
10535:
10529:
10523:
10518:
10514:
10510:
10505:
10501:
10497:
10494:
10489:
10483:
10477:
10472:
10468:
10464:
10459:
10455:
10451:
10448:
10445:
10443:
10439:
10433:
10427:
10422:
10418:
10414:
10409:
10405:
10401:
10398:
10393:
10387:
10381:
10376:
10372:
10368:
10363:
10359:
10355:
10352:
10351:
10348:
10345:
10340:
10334:
10326:
10322:
10318:
10313:
10307:
10299:
10295:
10291:
10286:
10280:
10272:
10268:
10264:
10259:
10253:
10245:
10241:
10237:
10234:
10231:
10229:
10227:
10222:
10216:
10208:
10204:
10200:
10195:
10189:
10181:
10177:
10173:
10168:
10162:
10154:
10150:
10146:
10141:
10135:
10127:
10123:
10119:
10116:
10115:
10095:
10090:
10084:
10078:
10073:
10069:
10065:
10060:
10056:
10052:
10049:
10044:
10038:
10032:
10027:
10023:
10019:
10014:
10010:
10006:
10003:
10000:
9995:
9989:
9981:
9977:
9973:
9968:
9962:
9954:
9950:
9946:
9943:
9940:
9935:
9929:
9921:
9917:
9913:
9908:
9902:
9894:
9890:
9886:
9874:
9871:
9856:
9855:
9852:
9849:
9780:
9769:
9762:
9755:
9744:
9737:
9730:
9707:
9702:
9697:
9692:
9687:
9682:
9679:
9677:
9673:
9669:
9665:
9664:
9659:
9654:
9649:
9644:
9639:
9634:
9631:
9629:
9625:
9621:
9617:
9616:
9611:
9606:
9601:
9596:
9591:
9586:
9583:
9581:
9577:
9573:
9569:
9568:
9563:
9558:
9553:
9548:
9543:
9538:
9535:
9533:
9529:
9525:
9521:
9520:
9515:
9510:
9505:
9500:
9495:
9490:
9487:
9485:
9481:
9477:
9473:
9472:
9467:
9462:
9457:
9452:
9447:
9442:
9439:
9437:
9433:
9429:
9425:
9424:
9419:
9414:
9409:
9404:
9399:
9394:
9391:
9389:
9385:
9381:
9377:
9376:
9371:
9366:
9361:
9356:
9351:
9346:
9343:
9341:
9337:
9333:
9329:
9328:
9323:
9318:
9313:
9308:
9303:
9298:
9295:
9293:
9289:
9285:
9281:
9280:
9261:refers to the
9252:
9243:
9230:
9177:
9172:
9166:
9163:
9162:
9159:
9156:
9155:
9152:
9149:
9148:
9146:
9139:
9131:
9127:
9123:
9119:
9115:
9111:
9107:
9103:
9099:
9098:
9093:
9089:
9085:
9081:
9077:
9073:
9069:
9065:
9061:
9060:
9055:
9051:
9047:
9043:
9039:
9035:
9031:
9027:
9023:
9022:
9020:
9015:
9010:
9004:
9001:
9000:
8997:
8994:
8993:
8990:
8987:
8986:
8984:
8954:
8951:
8946:
8942:
8938:
8935:
8930:
8926:
8922:
8919:
8914:
8910:
8906:
8903:
8901:
8899:
8896:
8895:
8892:
8889:
8884:
8880:
8876:
8873:
8868:
8864:
8860:
8857:
8852:
8848:
8844:
8841:
8839:
8837:
8834:
8833:
8830:
8827:
8822:
8818:
8814:
8811:
8806:
8802:
8798:
8795:
8790:
8786:
8782:
8779:
8777:
8775:
8772:
8771:
8743:
8738:
8733:
8728:
8723:
8718:
8713:
8710:
8705:
8700:
8695:
8690:
8685:
8680:
8677:
8672:
8667:
8662:
8657:
8652:
8647:
8644:
8641:
8639:
8637:
8634:
8633:
8630:
8625:
8620:
8615:
8610:
8605:
8600:
8597:
8592:
8587:
8582:
8577:
8572:
8567:
8564:
8559:
8554:
8549:
8544:
8539:
8534:
8531:
8528:
8526:
8524:
8521:
8520:
8517:
8512:
8507:
8502:
8497:
8492:
8487:
8484:
8479:
8474:
8469:
8464:
8459:
8454:
8451:
8446:
8441:
8436:
8431:
8426:
8421:
8418:
8415:
8413:
8411:
8408:
8407:
8379:
8374:
8369:
8364:
8361:
8356:
8351:
8346:
8343:
8338:
8333:
8328:
8325:
8320:
8315:
8310:
8307:
8304:
8301:
8299:
8297:
8294:
8293:
8290:
8285:
8280:
8275:
8272:
8267:
8262:
8257:
8254:
8249:
8244:
8239:
8236:
8231:
8226:
8221:
8218:
8215:
8212:
8210:
8208:
8205:
8204:
8201:
8196:
8191:
8186:
8183:
8178:
8173:
8168:
8165:
8160:
8155:
8150:
8147:
8142:
8137:
8132:
8129:
8126:
8123:
8121:
8119:
8116:
8115:
8047:The values of
8030:
8025:
8020:
8015:
8011:
8007:
8004:
8002:
8000:
7997:
7996:
7993:
7988:
7983:
7978:
7974:
7970:
7967:
7965:
7963:
7960:
7959:
7956:
7951:
7946:
7941:
7937:
7933:
7930:
7928:
7926:
7923:
7922:
7892:
7887:
7882:
7879:
7874:
7869:
7864:
7861:
7856:
7851:
7846:
7843:
7839:
7816:
7809:
7802:
7777:
7772:
7767:
7762:
7758:
7754:
7751:
7749:
7747:
7744:
7743:
7740:
7735:
7730:
7725:
7721:
7717:
7714:
7712:
7710:
7707:
7706:
7703:
7698:
7693:
7688:
7684:
7680:
7677:
7675:
7673:
7670:
7669:
7641:
7636:
7631:
7626:
7623:
7618:
7613:
7608:
7605:
7600:
7595:
7590:
7587:
7583:
7560:
7553:
7546:
7535:
7532:
7520:
7517:
7513:
7505:
7497:
7493:
7490:
7487:
7484:
7480:
7472:
7464:
7460:
7457:
7454:
7451:
7447:
7439:
7431:
7427:
7424:
7421:
7418:
7414:
7406:
7398:
7394:
7391:
7388:
7384:
7376:
7368:
7364:
7361:
7358:
7354:
7346:
7338:
7334:
7310:
7302:
7298:
7294:
7290:
7286:
7282:
7278:
7274:
7270:
7269:
7264:
7260:
7256:
7252:
7248:
7244:
7240:
7236:
7232:
7231:
7226:
7222:
7218:
7214:
7210:
7206:
7202:
7198:
7194:
7193:
7191:
7186:
7183:
7179:
7171:
7163:
7159:
7146:of the 3-by-3
7117:parallelepiped
7102:
7099:
7095:
7091:
7087:
7083:
7080:
7076:
7072:
7069:
7065:
7057:
7049:
7045:
7006:Main article:
7003:
7000:
6981:
6976:
6970:
6964:
6959:
6955:
6949:
6945:
6941:
6936:
6932:
6926:
6922:
6918:
6915:
6910:
6904:
6898:
6893:
6889:
6883:
6879:
6875:
6870:
6866:
6860:
6856:
6852:
6849:
6844:
6838:
6832:
6827:
6823:
6817:
6813:
6809:
6804:
6800:
6794:
6790:
6786:
6783:
6778:
6773:
6768:
6738:The length of
6618:
6613:
6610:
6607:
6604:
6601:
6597:
6593:
6589:
6584:
6580:
6576:
6572:
6568:
6564:
6560:
6522:vector product
6510:Main article:
6507:
6504:
6492:
6487:
6483:
6477:
6473:
6469:
6464:
6460:
6454:
6450:
6446:
6441:
6437:
6431:
6427:
6423:
6419:
6415:
6411:
6343:
6340:
6337:
6334:
6330:
6326:
6322:
6317:
6313:
6309:
6305:
6301:
6297:
6293:
6271:scalar product
6245:Main article:
6242:
6239:
6197:
6194:
6168:Main article:
6165:
6162:
6148:
6143:
6135:
6131:
6127:
6121:
6117:
6111:
6106:
6101:
6093:
6089:
6085:
6079:
6075:
6069:
6064:
6059:
6051:
6047:
6043:
6037:
6033:
6027:
6021:
6017:
6013:
6008:
6002:
5996:
5993:
5970:
5963:
5956:
5926:Main article:
5908:
5905:
5893:
5887:
5883:
5879:
5873:
5869:
5865:
5861:
5839:
5832:
5825:
5804:
5797:
5792:
5788:
5784:
5779:
5774:
5770:
5766:
5761:
5756:
5752:
5746:
5742:
5738:
5734:
5719:Euclidean norm
5707:absolute value
5693:of the vector
5672:
5669:
5543:
5538:
5533:
5528:
5523:
5519:
5515:
5512:
5509:
5504:
5499:
5494:
5489:
5485:
5481:
5478:
5475:
5470:
5465:
5460:
5455:
5451:
5447:
5444:
5441:
5437:
5433:
5381:Main article:
5378:
5375:
5289:
5284:
5279:
5274:
5269:
5265:
5261:
5256:
5252:
5248:
5245:
5240:
5235:
5230:
5225:
5221:
5217:
5212:
5208:
5204:
5201:
5196:
5191:
5186:
5181:
5177:
5173:
5168:
5164:
5160:
5157:
5153:
5149:
5145:
4974:
4969:
4964:
4959:
4954:
4950:
4946:
4941:
4937:
4933:
4930:
4925:
4920:
4915:
4910:
4906:
4902:
4897:
4893:
4889:
4886:
4881:
4876:
4871:
4866:
4862:
4858:
4853:
4849:
4845:
4842:
4838:
4834:
4830:
4803:
4800:
4770:
4765:
4761:
4757:
4754:
4749:
4745:
4740:
4735:
4731:
4727:
4724:
4719:
4715:
4710:
4705:
4701:
4697:
4694:
4689:
4685:
4658:
4652:
4644:
4640:
4636:
4631:
4625:
4617:
4613:
4609:
4604:
4598:
4590:
4586:
4582:
4577:
4549:
4543:
4535:
4531:
4527:
4522:
4516:
4508:
4504:
4500:
4495:
4489:
4481:
4477:
4473:
4468:
4444:
4441:
4428:
4423:
4419:
4415:
4410:
4406:
4401:
4396:
4392:
4388:
4383:
4379:
4374:
4369:
4365:
4361:
4356:
4352:
4329:
4323:
4315:
4311:
4307:
4302:
4296:
4288:
4284:
4280:
4275:
4269:
4261:
4257:
4253:
4248:
4224:
4218:
4210:
4206:
4202:
4197:
4191:
4183:
4179:
4175:
4170:
4164:
4156:
4152:
4148:
4143:
4129:
4126:
4114:
4109:
4103:
4095:
4091:
4087:
4082:
4076:
4068:
4064:
4060:
4055:
4049:
4041:
4037:
4033:
4028:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3981:
3976:
3970:
3961:
3958:
3955:
3952:
3949:
3946:
3943:
3940:
3937:
3932:
3926:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3888:
3882:
3857:
3854:
3781:
3778:
3772:
3766:
3763:
3757:
3751:
3748:
3718:
3715:
3709:
3703:
3700:
3694:
3688:
3685:
3651:
3648:
3642:
3636:
3633:
3627:
3621:
3618:
3563:
3560:
3552:index notation
3545:
3528:
3523:
3516:
3512:
3508:
3503:
3496:
3492:
3488:
3483:
3476:
3472:
3468:
3463:
3458:
3453:
3448:
3443:
3438:
3433:
3428:
3423:
3419:
3402:
3393:
3384:
3376:
3369:
3362:
3341:
3311:
3308:
3302:
3296:
3293:
3287:
3281:
3278:
3254:
3250:
3246:
3242:
3238:
3218:
3211:
3204:
3171:
3164:
3157:
3140:
3135:
3129:
3121:
3117:
3113:
3108:
3102:
3094:
3090:
3086:
3081:
3075:
3067:
3063:
3059:
3054:
3049:
3044:
3039:
3034:
3029:
3024:
3019:
3014:
3010:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2958:
2954:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2924:
2920:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2890:
2886:
2882:
2879:
2874:
2870:
2866:
2861:
2857:
2853:
2848:
2844:
2840:
2837:
2833:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2757:
2751:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2713:
2707:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2669:
2663:
2649:standard basis
2630:
2625:
2621:
2615:
2611:
2602:
2598:
2589:
2585:
2581:
2578:
2573:
2565:
2561:
2557:
2556:
2551:
2547:
2543:
2542:
2537:
2533:
2529:
2528:
2526:
2521:
2517:
2507:, as follows:
2482:
2479:
2474:
2470:
2466:
2461:
2458:
2455:
2451:
2447:
2444:
2441:
2436:
2432:
2428:
2423:
2419:
2415:
2410:
2406:
2402:
2399:
2395:
2362:
2359:
2354:
2350:
2346:
2341:
2337:
2333:
2328:
2324:
2320:
2317:
2313:
2303:also written,
2292:
2289:
2284:
2280:
2276:
2271:
2267:
2263:
2258:
2254:
2250:
2247:
2243:
2207:
2203:
2200:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2153:
2106:real numbers (
2035:terminal point
2021:is called the
2001:is called the
1977:
1937:
1932:
1929:
1885:
1881:
1878:
1843:
1840:
1802:
1779:
1756:
1718:
1715:
1651:
1648:
1644:thermodynamics
1568:
1565:
1553:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1373:
1369:
1366:
1333:
1330:
1322:magnetic field
1277:
1274:
1237:
1234:
1210:
1205:
1200:
1178:
1173:
1169:
1166:
1119:
1114:
1109:
1066:
1061:
1034:
1029:
1001:
996:
961:
958:
924:
919:
916:
903:additive group
886:
881:
878:
857:linear algebra
800:
795:
791:
788:
691:
688:
648:
641:
638:
633:
630:
601:
595:
592:
550:terminal point
480:
477:
382:imaginary part
368:imaginary unit
296:
293:
238:distributivity
222:multiplication
166:
159:
154:
151:
133:terminal point
82:spatial vector
26:
9:
6:
4:
3:
2:
13399:
13388:
13385:
13383:
13380:
13378:
13375:
13373:
13370:
13368:
13365:
13363:
13360:
13358:
13355:
13353:
13350:
13349:
13347:
13332:
13324:
13323:
13320:
13314:
13311:
13309:
13308:Sparse matrix
13306:
13304:
13301:
13299:
13296:
13294:
13291:
13290:
13288:
13286:
13282:
13276:
13273:
13271:
13268:
13266:
13263:
13261:
13258:
13256:
13253:
13251:
13248:
13247:
13245:
13243:constructions
13242:
13238:
13232:
13231:Outermorphism
13229:
13227:
13224:
13222:
13219:
13217:
13214:
13212:
13209:
13207:
13204:
13202:
13199:
13197:
13194:
13192:
13191:Cross product
13189:
13187:
13184:
13183:
13181:
13179:
13175:
13169:
13166:
13164:
13161:
13159:
13158:Outer product
13156:
13154:
13151:
13149:
13146:
13144:
13141:
13139:
13138:Orthogonality
13136:
13135:
13133:
13131:
13127:
13121:
13118:
13116:
13115:Cramer's rule
13113:
13111:
13108:
13106:
13103:
13101:
13098:
13096:
13093:
13091:
13088:
13086:
13085:Decomposition
13083:
13081:
13078:
13077:
13075:
13073:
13069:
13064:
13054:
13051:
13049:
13046:
13044:
13041:
13039:
13036:
13034:
13031:
13029:
13026:
13024:
13021:
13019:
13016:
13014:
13011:
13009:
13006:
13004:
13001:
12999:
12996:
12994:
12991:
12989:
12986:
12984:
12981:
12979:
12976:
12974:
12971:
12969:
12966:
12964:
12961:
12960:
12958:
12954:
12948:
12945:
12943:
12940:
12939:
12936:
12932:
12925:
12920:
12918:
12913:
12911:
12906:
12905:
12902:
12895:
12891:
12888:
12885:
12881:
12878:
12874:
12870:
12869:
12864:
12860:
12859:
12849:
12843:
12839:
12838:
12833:
12829:
12825:
12819:
12815:
12814:
12808:
12807:
12797:
12795:0-486-65812-0
12791:
12786:
12785:
12779:
12778:Pedoe, Daniel
12775:
12771:
12769:0-387-96205-0
12765:
12761:
12757:
12753:
12748:
12743:
12739:
12735:
12734:
12729:
12724:
12720:
12714:
12710:
12706:
12701:
12697:
12695:1-55369-133-4
12691:
12687:
12686:
12680:
12676:
12670:
12665:
12664:
12658:
12654:
12650:
12644:
12640:
12639:
12634:
12630:
12629:
12611:
12607:
12603:
12601:9781563479274
12597:
12593:
12586:
12579:
12574:
12572:
12570:
12568:
12559:
12557:0-387-94746-9
12553:
12549:
12548:
12540:
12538:
12536:
12521:on 2007-01-22
12520:
12516:
12510:
12502:
12496:
12490:
12484:
12478:
12473:
12469:
12464:
12458:
12453:
12438:
12434:
12428:
12426:
12418:
12414:
12408:
12404:
12396:
12389:
12385:
12379:
12372:
12367:
12360:
12354:
12335:
12328:
12323:
12319:
12314:
12312:
12310:
12308:
12293:
12289:
12282:
12268:
12264:
12258:
12256:
12254:
12252:
12237:
12233:
12227:
12225:
12216:
12214:9780195219425
12210:
12206:
12200:
12186:
12183:Jeff Miller.
12178:
12170:
12166:
12165:
12159:
12157:
12149:
12145:
12141:
12135:
12129:
12125:
12120:
12114:
12109:
12103:
12098:
12094:
12083:
12079:
12068:
12065:
12063:
12060:
12058:
12055:
12053:
12052:Vector bundle
12050:
12048:
12045:
12043:
12040:
12038:(of a vector)
12037:
12034:
12032:
12029:
12027:
12024:
12022:
12019:
12017:
12014:
12012:
12009:
12007:
12006:Normal vector
12004:
12002:
12001:Hilbert space
11999:
11997:
11993:
11990:
11988:
11985:
11983:
11979:
11976:
11974:
11971:
11969:
11966:
11964:
11961:
11959:
11956:
11954:
11951:
11949:
11945:
11942:
11941:
11934:
11932:
11927:
11925:
11921:
11915:
11913:
11909:
11905:
11901:
11896:
11894:
11893:cross product
11890:
11889:polar vectors
11886:
11882:
11878:
11877:
11872:
11871:
11864:
11861:
11859:
11855:
11851:
11850:contravariant
11847:
11843:
11839:
11834:
11832:
11828:
11824:
11820:
11816:
11812:
11808:
11803:
11799:
11795:
11791:
11787:
11783:
11778:
11761:
11758:
11755:
11747:
11743:
11738:
11733:
11730:
11726:
11721:
11717:
11714:
11710:
11706:
11702:
11691:
11688:
11673:
11670:
11662:
11659:December 2021
11652:
11648:
11642:
11639:This section
11637:
11628:
11627:
11618:
11615:
11607:
11604:December 2021
11597:
11593:
11589:
11583:
11582:
11578:
11573:This section
11571:
11567:
11562:
11561:
11556:
11554:
11547:
11546:
11541:
11540:
11535:
11530:
11521:
11520:
11512:
11499:
11491:
11479:
11474:
11459:
11449:
11446:
11439:
11435:
11430:
11410:
11407:
11392:
11388:
11379:
11377:
11373:
11370:
11366:
11364:
11360:
11353:
11337:
11332:
11320:
11310:
11307:
11302:
11285:
11281:
11277:
11274:
11269:
11267:
11263:
11259:
11255:
11239:
11234:
11217:
11213:
11209:
11204:
11200:
11193:
11188:
11171:
11167:
11163:
11158:
11154:
11147:
11142:
11125:
11121:
11117:
11112:
11108:
11101:
11091:
11076:
11069:
11062:
11055:
11051:
11044:
11037:
11030:
11026:
11021:
11019:
11003:
10998:
10984:
10980:
10976:
10971:
10957:
10953:
10949:
10944:
10930:
10926:
10922:
10907:
10900:
10893:
10886:
10882:
10872:
10870:
10866:
10862:
10858:
10854:
10850:
10846:
10842:
10836:
10826:
10824:
10819:
10815:
10812:
10811:dimensionless
10802:
10798:
10788:
10786:
10781:
10768:
10763:
10751:
10736:
10732:
10726:
10722:
10718:
10713:
10709:
10703:
10699:
10692:
10684:
10670:
10666:
10662:
10657:
10643:
10639:
10632:
10624:
10610:
10606:
10602:
10597:
10583:
10579:
10568:
10564:
10560:
10555:
10538:
10533:
10516:
10512:
10508:
10503:
10499:
10492:
10487:
10470:
10466:
10462:
10457:
10453:
10446:
10444:
10437:
10420:
10416:
10412:
10407:
10403:
10396:
10391:
10374:
10370:
10366:
10361:
10357:
10346:
10338:
10324:
10320:
10316:
10311:
10297:
10293:
10289:
10284:
10270:
10266:
10262:
10257:
10243:
10239:
10232:
10230:
10220:
10206:
10202:
10198:
10193:
10179:
10175:
10171:
10166:
10152:
10148:
10144:
10139:
10125:
10121:
10093:
10088:
10071:
10067:
10063:
10058:
10054:
10047:
10042:
10025:
10021:
10017:
10012:
10008:
10001:
9993:
9979:
9975:
9971:
9966:
9952:
9948:
9941:
9933:
9919:
9915:
9911:
9906:
9892:
9888:
9870:
9867:
9865:
9861:
9853:
9850:
9847:
9846:
9845:
9842:
9840:
9836:
9832:
9828:
9824:
9820:
9816:
9812:
9808:
9804:
9800:
9796:
9792:
9788:
9783:
9779:
9775:
9768:
9761:
9754:
9751:basis and to
9750:
9743:
9736:
9729:
9724:
9705:
9695:
9690:
9680:
9678:
9671:
9667:
9657:
9647:
9642:
9632:
9630:
9623:
9619:
9609:
9599:
9594:
9584:
9582:
9575:
9571:
9561:
9551:
9546:
9536:
9534:
9527:
9523:
9513:
9503:
9498:
9488:
9486:
9479:
9475:
9465:
9455:
9450:
9440:
9438:
9431:
9427:
9417:
9407:
9402:
9392:
9390:
9383:
9379:
9369:
9359:
9354:
9344:
9342:
9335:
9331:
9321:
9311:
9306:
9296:
9294:
9287:
9283:
9270:
9269:. Therefore,
9268:
9264:
9260:
9255:
9251:
9246:
9242:
9238:
9233:
9229:
9225:
9221:
9217:
9213:
9209:
9205:
9201:
9197:
9193:
9188:
9175:
9170:
9164:
9157:
9150:
9144:
9137:
9129:
9125:
9117:
9113:
9105:
9101:
9091:
9087:
9079:
9075:
9067:
9063:
9053:
9049:
9041:
9037:
9029:
9025:
9018:
9013:
9008:
9002:
8995:
8988:
8982:
8972:
8969:
8952:
8949:
8944:
8940:
8936:
8933:
8928:
8924:
8920:
8917:
8912:
8908:
8904:
8902:
8897:
8890:
8887:
8882:
8878:
8874:
8871:
8866:
8862:
8858:
8855:
8850:
8846:
8842:
8840:
8835:
8828:
8825:
8820:
8816:
8812:
8809:
8804:
8800:
8796:
8793:
8788:
8784:
8780:
8778:
8773:
8761:
8758:
8741:
8736:
8726:
8721:
8711:
8708:
8703:
8693:
8688:
8678:
8675:
8670:
8660:
8655:
8645:
8642:
8640:
8635:
8628:
8623:
8613:
8608:
8598:
8595:
8590:
8580:
8575:
8565:
8562:
8557:
8547:
8542:
8532:
8529:
8527:
8522:
8515:
8510:
8500:
8495:
8485:
8482:
8477:
8467:
8462:
8452:
8449:
8444:
8434:
8429:
8419:
8416:
8414:
8409:
8397:
8394:
8377:
8372:
8362:
8354:
8344:
8341:
8336:
8326:
8323:
8318:
8308:
8302:
8300:
8295:
8288:
8283:
8273:
8265:
8255:
8252:
8247:
8237:
8234:
8229:
8219:
8213:
8211:
8206:
8199:
8194:
8184:
8176:
8166:
8163:
8158:
8148:
8145:
8140:
8130:
8124:
8122:
8117:
8105:
8103:
8099:
8095:
8091:
8087:
8083:
8079:
8074:
8070:
8066:
8062:
8058:
8054:
8050:
8045:
8028:
8023:
8013:
8005:
8003:
7998:
7991:
7986:
7976:
7968:
7966:
7961:
7954:
7949:
7939:
7931:
7929:
7924:
7912:
7910:
7905:
7890:
7880:
7877:
7872:
7862:
7859:
7854:
7844:
7841:
7828:
7826:
7823:, the vector
7822:
7815:
7808:
7801:
7797:
7792:
7775:
7770:
7760:
7752:
7750:
7745:
7738:
7733:
7723:
7715:
7713:
7708:
7701:
7696:
7686:
7678:
7676:
7671:
7659:
7657:
7652:
7639:
7634:
7624:
7621:
7616:
7606:
7603:
7598:
7588:
7585:
7572:
7570:
7566:
7559:
7552:
7545:
7541:
7531:
7518:
7488:
7485:
7455:
7452:
7422:
7419:
7389:
7359:
7323:
7308:
7300:
7296:
7288:
7284:
7276:
7272:
7262:
7258:
7250:
7246:
7238:
7234:
7224:
7220:
7212:
7208:
7200:
7196:
7189:
7184:
7149:
7145:
7141:
7136:
7134:
7130:
7126:
7122:
7118:
7113:
7100:
7089:
7078:
7070:
7035:
7033:
7030:
7027:
7023:
7019:
7015:
7009:
6999:
6997:
6992:
6979:
6974:
6957:
6953:
6947:
6943:
6939:
6934:
6930:
6924:
6920:
6913:
6908:
6891:
6887:
6881:
6877:
6873:
6868:
6864:
6858:
6854:
6847:
6842:
6825:
6821:
6815:
6811:
6807:
6802:
6798:
6792:
6788:
6781:
6771:
6755:
6753:
6749:
6745:
6742: ×
6741:
6736:
6734:
6730:
6726:
6722:
6718:
6715: ×
6714:
6710:
6706:
6702:
6699: ×
6698:
6689:
6685:
6683:
6679:
6675:
6671:
6667:
6663:
6659:
6655:
6651:
6650:perpendicular
6647:
6643:
6639:
6635:
6630:
6608:
6602:
6599:
6570:
6562:
6549:
6547:
6543:
6539:
6536: ×
6535:
6531:
6527:
6526:outer product
6523:
6519:
6518:cross product
6513:
6512:Cross product
6506:Cross product
6503:
6490:
6485:
6481:
6475:
6471:
6467:
6462:
6458:
6452:
6448:
6444:
6439:
6435:
6429:
6425:
6421:
6413:
6400:
6397:
6395:
6391:
6387:
6383:
6379:
6375:
6371:
6367:
6363:
6359:
6354:
6341:
6338:
6335:
6332:
6303:
6295:
6282:
6280:
6277: ∙
6276:
6272:
6268:
6267:
6266:inner product
6262:
6258:
6254:
6248:
6238:
6235:
6231:
6227:
6222:
6218:
6214:
6192:
6177:
6171:
6161:
6146:
6119:
6115:
6109:
6104:
6077:
6073:
6067:
6062:
6035:
6031:
6025:
6000:
5980:
5978:
5969:
5962:
5955:
5951:
5945:
5943:
5939:
5935:
5929:
5922:
5918:
5913:
5904:
5891:
5881:
5871:
5850:
5848:
5843:
5838:
5831:
5824:
5820:
5815:
5802:
5795:
5790:
5786:
5782:
5777:
5772:
5768:
5764:
5759:
5754:
5750:
5744:
5723:
5721:
5720:
5715:
5710:
5708:
5704:
5700:
5696:
5692:
5691:
5686:
5685:
5680:
5679:
5668:
5666:
5662:
5658:
5654:
5650:
5646:
5642:
5638:
5635:
5631:
5628:
5624:
5620:
5616:
5612:
5605:
5601:
5597:
5592:
5588:
5586:
5582:
5578:
5573:
5571:
5567:
5563:
5559:
5554:
5541:
5536:
5521:
5517:
5513:
5507:
5502:
5487:
5483:
5479:
5473:
5468:
5453:
5449:
5445:
5439:
5431:
5423:
5421:
5417:
5413:
5409:
5405:
5402:
5398:
5389:
5384:
5372:
5368:
5363:
5359:
5357:
5353:
5349:
5345:
5341:
5337:
5333:
5329:
5325:
5321:
5317:
5313:
5309:
5305:
5300:
5287:
5282:
5267:
5263:
5259:
5254:
5250:
5243:
5238:
5223:
5219:
5215:
5210:
5206:
5199:
5194:
5179:
5175:
5171:
5166:
5162:
5155:
5147:
5134:
5132:
5128:
5123:
5121:
5117:
5113:
5109:
5105:
5101:
5097:
5093:
5089:
5085:
5081:
5077:
5073:
5069:
5065:
5061:
5057:
5056:parallelogram
5053:
5049:
5045:
5038:
5034:
5029:
5025:
5023:
5019:
5015:
5011:
5007:
5003:
4998:
4996:
4992:
4988:
4972:
4967:
4952:
4948:
4944:
4939:
4935:
4928:
4923:
4908:
4904:
4900:
4895:
4891:
4884:
4879:
4864:
4860:
4856:
4851:
4847:
4840:
4832:
4819:
4815:
4809:
4799:
4797:
4793:
4792:codirectional
4789:
4788:
4782:
4768:
4763:
4759:
4755:
4752:
4747:
4743:
4738:
4733:
4729:
4725:
4722:
4717:
4713:
4708:
4703:
4699:
4695:
4692:
4687:
4683:
4674:
4671:
4656:
4642:
4638:
4634:
4629:
4615:
4611:
4607:
4602:
4588:
4584:
4580:
4565:
4562:
4547:
4533:
4529:
4525:
4520:
4506:
4502:
4498:
4493:
4479:
4475:
4471:
4456:
4454:
4450:
4440:
4426:
4421:
4417:
4413:
4408:
4404:
4399:
4394:
4390:
4386:
4381:
4377:
4372:
4367:
4363:
4359:
4354:
4350:
4342:are equal if
4327:
4313:
4309:
4305:
4300:
4286:
4282:
4278:
4273:
4259:
4255:
4251:
4222:
4208:
4204:
4200:
4195:
4181:
4177:
4173:
4168:
4154:
4150:
4146:
4125:
4112:
4107:
4093:
4089:
4085:
4080:
4066:
4062:
4058:
4053:
4039:
4035:
4031:
4016:
3997:
3994:
3991:
3988:
3985:
3979:
3974:
3959:
3953:
3950:
3947:
3944:
3941:
3935:
3930:
3915:
3909:
3906:
3903:
3900:
3897:
3891:
3886:
3869:
3863:
3853:
3851:
3847:
3842:
3840:
3836:
3832:
3831:
3827:
3822:
3821:
3816:
3812:
3808:
3804:
3799:
3796:
3770:
3755:
3735:
3707:
3692:
3672:
3668:
3640:
3625:
3604:
3597:
3593:
3591:
3587:
3583:
3579:
3575:
3572:As explained
3569:
3559:
3557:
3553:
3548:
3544:
3541:The notation
3539:
3526:
3514:
3510:
3506:
3494:
3490:
3486:
3474:
3470:
3466:
3461:
3451:
3446:
3436:
3431:
3421:
3408:
3405:
3401:
3396:
3392:
3387:
3383:
3379:
3372:
3365:
3358:
3328:
3300:
3285:
3248:
3240:
3226:
3224:
3217:
3210:
3203:
3199:
3195:
3191:
3187:
3183:
3179:
3178:
3170:
3163:
3156:
3151:
3138:
3133:
3119:
3115:
3111:
3106:
3092:
3088:
3084:
3079:
3065:
3061:
3057:
3052:
3042:
3037:
3027:
3022:
3012:
2998:
2982:
2976:
2973:
2970:
2967:
2964:
2956:
2952:
2948:
2942:
2939:
2936:
2933:
2930:
2922:
2918:
2914:
2908:
2905:
2902:
2899:
2896:
2888:
2884:
2880:
2872:
2868:
2864:
2859:
2855:
2851:
2846:
2842:
2835:
2821:
2816:
2812:
2808:
2804:
2800:
2784:
2778:
2775:
2772:
2769:
2766:
2760:
2755:
2740:
2734:
2731:
2728:
2725:
2722:
2716:
2711:
2696:
2690:
2687:
2684:
2681:
2678:
2672:
2667:
2650:
2646:
2641:
2628:
2613:
2609:
2600:
2596:
2587:
2583:
2576:
2571:
2563:
2559:
2549:
2545:
2535:
2531:
2524:
2519:
2506:
2502:
2498:
2497:column vector
2493:
2480:
2472:
2468:
2464:
2459:
2456:
2453:
2449:
2445:
2442:
2439:
2434:
2430:
2426:
2421:
2417:
2413:
2408:
2404:
2397:
2383:
2378:
2377:n-dimensional
2373:
2360:
2352:
2348:
2344:
2339:
2335:
2331:
2326:
2322:
2315:
2290:
2282:
2278:
2274:
2269:
2265:
2261:
2256:
2252:
2245:
2231:
2226:
2221:
2205:
2201:
2198:
2186:
2173:
2167:
2164:
2161:
2155:
2142:
2138:
2133:
2131:
2127:
2126:
2121:
2117:
2113:
2109:
2105:
2101:
2097:
2093:
2085:
2078:
2073:
2069:
2067:
2062:
2058:
2057:perpendicular
2050:
2046:
2044:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2015:initial point
2012:
2008:
2004:
2000:
1996:
1995:line segments
1991:
1965:
1961:
1957:
1956:
1930:
1927:
1915:
1911:
1908:from a point
1907:
1903:
1883:
1880:
1876:
1866:
1862:
1861:
1838:
1827:
1823:
1819:
1818:
1791:
1745:
1738:
1734:
1729:
1724:
1714:
1712:
1708:
1704:
1700:
1695:
1693:
1689:
1685:
1681:
1677:
1673:
1669:
1668:contravariant
1665:
1661:
1660:basis vectors
1657:
1647:
1645:
1641:
1640:affine spaces
1637:
1636:vector spaces
1632:
1630:
1626:
1622:
1617:
1615:
1611:
1610:parallelotope
1608:-dimensional
1607:
1603:
1599:
1598:parallelogram
1595:
1591:
1587:
1586:cross product
1582:
1578:
1574:
1564:
1551:
1544:
1541:
1538:
1535:
1532:
1529:
1523:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1481:
1475:
1472:
1469:
1466:
1463:
1460:
1454:
1448:
1445:
1442:
1439:
1436:
1423:
1421:
1415:
1409:
1407:
1401:
1396:
1390:
1371:
1367:
1364:
1351:
1344:
1339:
1329:
1327:
1323:
1319:
1315:
1311:
1307:
1303:
1299:
1295:
1291:
1287:
1283:
1273:
1271:
1267:
1263:
1259:
1255:
1251:
1247:
1243:
1233:
1231:
1227:
1208:
1203:
1176:
1171:
1167:
1164:
1150:of any point
1149:
1145:
1141:
1137:
1117:
1112:
1097:
1086:of dimension
1085:
1082:
1064:
1048:
1032:
1017:
999:
985:
979:
977:
959:
956:
946:
942:
938:
922:
917:
914:
904:
900:
884:
879:
876:
866:
858:
853:
851:
847:
843:
839:
825:
821:
816:
811:
798:
793:
789:
786:
773:
769:
763:
759:
758:parallelogram
754:
750:
746:
742:
735:
731:
723:
719:
713:
712:ordered pairs
709:
705:
701:
697:
694:In classical
687:
685:
680:
678:
674:
673:parallelogram
670:
666:
639:
636:
631:
628:
593:
590:
577:
575:
571:
567:
563:
562:
557:
556:
551:
547:
546:initial point
542:
540:
539:
534:
533:
528:
527:
522:
518:
514:
510:
506:
502:
498:
494:
490:
486:
476:
474:
473:
468:
464:
460:
456:
452:
448:
446:
445:cross product
442:
438:
434:
433:
428:
424:
420:
416:
412:
408:
404:
403:August Möbius
400:
396:
390:
385:
383:
378:
373:
369:
365:
361:
357:
353:
350:(also called
348:
344:
339:
335:
331:
326:
323:as part of a
322:
318:
314:
310:
306:
302:
292:
290:
286:
285:pseudovectors
282:
278:
274:
270:
266:
262:
258:
254:
249:
247:
243:
239:
235:
234:associativity
231:
230:commutativity
227:
223:
219:
215:
211:
207:
203:
199:
195:
190:
186:
183:to the point
182:
177:
164:
152:
149:
137:
134:
130:
127:
126:initial point
123:
122:
117:
113:
109:
105:
104:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
59:
52:
48:
43:
37:
33:
19:
13241:Vector space
12973:Vector space
12967:
12866:
12836:
12812:
12783:
12759:
12746:
12731:
12704:
12684:
12662:
12657:Apostol, Tom
12637:
12633:Apostol, Tom
12591:
12585:
12546:
12523:. Retrieved
12519:the original
12509:
12500:
12494:
12488:
12482:
12476:
12471:
12463:
12452:
12441:. Retrieved
12439:. 2013-11-07
12436:
12416:
12413:bound vector
12412:
12411:is called a
12406:
12402:
12395:
12390:, p. 9.
12383:
12378:
12366:
12358:
12353:
12341:. Retrieved
12334:the original
12295:. Retrieved
12291:
12281:
12270:. Retrieved
12266:
12239:. Retrieved
12235:
12204:
12199:
12188:. Retrieved
12162:
12155:
12147:
12143:
12134:
12119:
12108:
12097:
12082:
12026:Pseudovector
11995:
11953:Banach space
11944:Affine space
11933:properties.
11928:
11911:
11897:
11888:
11885:true vectors
11884:
11881:axial vector
11880:
11876:pseudovector
11874:
11868:
11862:
11841:
11835:
11831:acceleration
11811:displacement
11801:
11793:
11789:
11785:
11781:
11776:
11745:
11741:
11736:
11731:
11728:
11724:
11719:
11715:
11704:
11698:
11683:
11665:
11656:
11640:
11610:
11601:
11586:Please help
11574:
11550:
11543:
11537:
11536:Please help
11533:
11438:displacement
11431:
11385:
11371:
11369:Acceleration
11367:
11358:
11351:
11283:
11275:
11270:
11265:
11261:
11257:
11256:relative to
11253:
11077:is a vector
11075:displacement
11067:
11060:
11053:
11049:
11042:
11035:
11028:
11024:
11022:
10898:
10891:
10884:
10880:
10878:
10852:
10848:
10844:
10840:
10838:
10808:
10800:
10782:
10567:pseudoscalar
10556:
9876:
9868:
9860:Euler angles
9857:
9843:
9838:
9834:
9830:
9826:
9814:
9810:
9806:
9802:
9794:
9790:
9781:
9777:
9773:
9766:
9759:
9752:
9748:
9741:
9734:
9727:
9725:
9271:
9258:
9253:
9249:
9244:
9240:
9231:
9227:
9223:
9219:
9215:
9211:
9207:
9203:
9199:
9195:
9191:
9189:
8973:
8970:
8762:
8759:
8398:
8395:
8106:
8097:
8093:
8089:
8088:in terms of
8085:
8081:
8077:
8072:
8068:
8064:
8060:
8056:
8052:
8048:
8046:
7913:
7908:
7906:
7829:
7824:
7820:
7813:
7806:
7799:
7795:
7793:
7660:
7655:
7653:
7573:
7568:
7564:
7557:
7550:
7543:
7539:
7537:
7324:
7139:
7137:
7132:
7128:
7124:
7114:
7036:
7031:
7028:
7025:
7021:
7017:
7013:
7011:
6996:pseudovector
6993:
6756:
6751:
6747:
6743:
6739:
6737:
6724:
6720:
6716:
6712:
6708:
6704:
6700:
6696:
6694:
6681:
6677:
6673:
6669:
6665:
6662:right-handed
6657:
6653:
6645:
6641:
6637:
6633:
6631:
6550:
6545:
6541:
6537:
6533:
6525:
6521:
6517:
6515:
6401:
6398:
6393:
6389:
6385:
6381:
6377:
6369:
6365:
6357:
6355:
6283:
6278:
6274:
6270:
6264:
6260:
6256:
6252:
6250:
6233:
6229:
6225:
6220:
6216:
6212:
6175:
6173:
5981:
5979:‖. That is:
5976:
5967:
5960:
5953:
5949:
5946:
5941:
5937:
5933:
5931:
5920:
5916:
5851:
5844:
5836:
5829:
5822:
5816:
5724:
5717:
5713:
5711:
5702:
5698:
5694:
5688:
5682:
5676:
5674:
5664:
5660:
5656:
5652:
5648:
5644:
5640:
5636:
5633:
5629:
5626:
5622:
5618:
5614:
5611:distributive
5608:
5603:
5602:of a vector
5599:
5595:
5584:
5580:
5576:
5574:
5569:
5565:
5561:
5557:
5555:
5424:
5419:
5415:
5411:
5403:
5396:
5394:
5370:
5366:
5355:
5351:
5347:
5343:
5339:
5335:
5331:
5327:
5323:
5319:
5315:
5311:
5307:
5303:
5301:
5135:
5130:
5126:
5124:
5119:
5115:
5111:
5107:
5103:
5099:
5095:
5091:
5087:
5083:
5079:
5075:
5071:
5067:
5063:
5059:
5051:
5047:
5043:
5041:
5036:
5032:
5021:
5017:
5013:
5009:
5005:
5001:
4999:
4994:
4990:
4986:
4817:
4813:
4811:
4808:Vector space
4795:
4791:
4785:
4783:
4675:
4672:
4566:
4563:
4457:
4448:
4446:
4131:
4014:
3865:
3845:
3843:
3824:
3818:
3814:
3810:
3806:
3800:
3797:
3605:
3602:
3589:
3585:
3571:
3546:
3542:
3540:
3409:
3403:
3399:
3394:
3390:
3385:
3381:
3374:
3367:
3360:
3357:unit vectors
3266:instead (or
3227:
3215:
3208:
3201:
3197:
3193:
3189:
3185:
3181:
3175:
3168:
3161:
3154:
3152:
2999:
2819:
2814:
2806:
2802:
2798:
2644:
2642:
2494:
2381:
2376:
2374:
2229:
2224:
2222:
2187:
2140:
2136:
2134:
2129:
2123:
2107:
2103:
2091:
2089:
2076:
2054:
2038:
2034:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
2002:
1998:
1992:
1954:
1953:
1913:
1909:
1906:displacement
1859:
1858:
1816:
1815:
1768:
1741:
1736:
1732:
1703:vector space
1696:
1667:
1663:
1653:
1633:
1618:
1613:
1605:
1576:
1572:
1570:
1424:
1419:
1413:
1410:
1405:
1399:
1394:
1388:
1349:
1342:
1335:
1326:vector field
1302:displacement
1293:
1289:
1279:
1265:
1261:
1257:
1245:
1239:
1080:
1049:
980:
976:translations
945:Affine space
899:group action
854:
850:acceleration
823:
819:
815:line segment
812:
771:
767:
761:
752:
748:
744:
740:
733:
729:
721:
717:
708:equipollence
693:
683:
681:
668:
578:
560:
559:
555:bound vector
554:
553:
549:
545:
543:
537:
536:
531:
530:
525:
524:
517:vector space
482:
470:
462:
458:
449:
430:
429:∇. In 1878,
427:del operator
422:
414:
392:
387:
381:
376:
355:
351:
346:
337:
333:
329:
316:
305:equipollence
298:
273:displacement
261:acceleration
250:
246:vector space
210:real numbers
201:
197:
194:displacement
188:
184:
180:
178:
135:
132:
128:
125:
119:
110:, including
101:
98:vector space
81:
77:
73:
72:or simply a
69:
55:
50:
46:
13221:Multivector
13186:Determinant
13143:Dot product
12988:Linear span
12756:Lang, Serge
12468:Gibbs, J.W.
12417:free vector
12102:Ivanov 2001
12047:Unit vector
12011:Null vector
11978:Four-vector
11870:orientation
9797:", or the "
9267:dot product
9257:. The term
7144:determinant
7018:box product
6253:dot product
6247:Dot product
6241:Dot product
6176:zero vector
6170:Zero vector
6164:Zero vector
5938:normalizing
5934:unit vector
5928:Unit vector
5907:Unit vector
5847:dot product
5401:real number
4812:The sum of
3803:orientation
3582:projections
2809:-axis of a
2116:coordinates
2039:final point
1912:to a point
1699:mathematics
1612:defined by
1594:orientation
1581:dot product
1416:= (0, 0, 0)
1352:= (0, 1, 0)
1345:= (1, 0, 0)
1016:dot product
974:are called
665:equipollent
579:Two arrows
561:free vector
489:engineering
441:dot product
389:quaternion.
343:real number
218:subtraction
66:engineering
58:mathematics
13352:Kinematics
13346:Categories
13255:Direct sum
13090:Invertible
12993:Linear map
12621:References
12525:2007-01-05
12443:2020-08-19
12343:2010-09-04
12297:2020-08-19
12272:2020-08-19
12241:2020-08-19
12190:2007-05-25
12179:required.)
12128:Pedoe 1988
12031:Quaternion
11982:relativity
11912:reflection
11858:chain rule
11539:improve it
10865:integrated
10857:parametric
10855:) gives a
9864:quaternion
6754:as sides.
6729:orthogonal
6676:, namely,
6223:(that is,
3860:See also:
3839:orthogonal
3592:that set.
3586:decomposed
3327:hat symbol
2501:row vector
1705:over some
1096:isomorphic
941:transitive
469:published
380:to be the
325:quaternion
13285:Numerical
13048:Transpose
12873:EMS Press
12816:. Dover.
12788:. Dover.
12738:EMS Press
12709:MIT Press
12610:652389481
12388:Lang 1986
12263:"Vectors"
11992:Grassmann
11805:abstract
11759:−
11575:does not
11545:talk page
11480:−
11460:⋅
11210:−
11164:−
11118:−
11092:−
10719:−
10633:∧
9823:transpose
9696:⋅
9648:⋅
9600:⋅
9552:⋅
9504:⋅
9456:⋅
9408:⋅
9360:⋅
9312:⋅
9239:relating
8727:⋅
8694:⋅
8661:⋅
8614:⋅
8581:⋅
8548:⋅
8501:⋅
8468:⋅
8435:⋅
8363:⋅
8274:⋅
8185:⋅
8014:⋅
7977:⋅
7940:⋅
7761:⋅
7724:⋅
7687:⋅
7489:−
7456:−
7423:−
7090:×
7079:⋅
6940:−
6874:−
6808:−
6772:×
6609:θ
6603:
6563:×
6414:⋅
6339:θ
6336:
6296:⋅
6196:→
6180:(0, 0, 0)
5995:^
5882:⋅
5684:magnitude
5583:= −1 and
5399:, by any
5260:−
5216:−
5172:−
5148:−
4756:−
4726:−
4696:−
3780:^
3777:ϕ
3765:^
3762:θ
3750:^
3717:^
3702:^
3699:ϕ
3687:^
3684:ρ
3650:^
3635:^
3620:^
3340:^
3310:^
3295:^
3280:^
2457:−
2443:⋯
2206:→
2043:magnitude
1936:⟶
1884:∼
1842:→
1822:Uppercase
1744:lowercase
1664:covariant
1616:vectors.
1530:−
1491:−
1461:−
1372:→
1172:→
960:→
935:which is
918:→
880:→
794:→
682:The term
647:⟶
600:⟶
541:vectors.
538:Euclidean
526:geometric
493:magnitude
372:real line
242:Euclidean
158:⟶
94:direction
86:magnitude
13331:Category
13270:Subspace
13265:Quotient
13216:Bivector
13130:Bilinear
13072:Matrices
12947:Glossary
12863:"Vector"
12780:(1988).
12758:(1986).
12728:"Vector"
12663:Calculus
12659:(1969).
12638:Calculus
12635:(1967).
12470:(1901).
12371:Itô 1993
12288:"Vector"
12154:"vector
12124:Itô 1993
11937:See also
11931:symmetry
11844:to be a
11823:momentum
11815:velocity
11798:velocity
11286:will be
11273:velocity
11073:) their
10869:calculus
10563:bivector
6652:to both
6596:‖
6588:‖
6583:‖
6575:‖
6364:between
6329:‖
6321:‖
6316:‖
6308:‖
6134:‖
6126:‖
6092:‖
6084:‖
6050:‖
6042:‖
6020:‖
6012:‖
5868:‖
5860:‖
5741:‖
5733:‖
5046:because
4796:parallel
4449:opposite
4128:Equality
3835:parallel
3830:rotation
3554:and the
2505:matrices
2031:endpoint
1902:distance
1826:matrices
1697:In pure
1672:gradient
1318:electric
1282:velocity
842:velocity
640:′
632:′
479:Overview
309:parallel
269:position
257:velocity
226:negation
214:addition
212:such as
12942:Outline
12875:, 2001
11800:, then
11645:Please
11596:removed
11581:sources
10791:Physics
9819:inverse
9772:as the
9747:as the
9235:is the
9214:basis (
9198:basis (
9194:in the
7542:basis {
5412:scalars
5334:, with
3841:to it.
3811:tangent
2805:-, and
2059:to the
1964:fraktur
1688:Tensors
1422:-axis.
1408:-axis.
1404:on the
1393:on the
1336:In the
1250:newtons
901:of the
838:scalars
698:(i.e.,
532:spatial
485:physics
366:use an
360:classes
295:History
289:tensors
253:physics
204:. Many
131:with a
116:support
62:physics
13226:Tensor
13038:Kernel
12968:Vector
12963:Scalar
12844:
12820:
12792:
12766:
12715:
12692:
12671:
12645:
12608:
12598:
12554:
12386:. See
12211:
12150:, see
12148:vector
12042:Tensor
11948:points
11924:torque
11916:actual
11908:wheels
11879:or an
11846:tensor
11829:, and
11807:vector
11792:, and
11350:where
11018:length
9263:cosine
9222:, and
9206:, and
8059:, and
7508:
7500:
7475:
7467:
7442:
7434:
7409:
7401:
7379:
7371:
7349:
7341:
7174:
7166:
7148:matrix
7060:
7052:
6711:, and
6680:and (−
6644:, and
6632:where
6356:where
5678:length
5671:Length
5663:+ (−1)
5414:(from
5397:scaled
3963:
3919:
3846:global
3826:radius
3815:radial
3809:, and
3807:normal
3578:add up
3380:, and
3196:, and
3153:where
2986:
2744:
2700:
2606:
2593:
2003:origin
1960:German
1692:tensor
1573:length
1312:, and
1270:meters
1248:of 15
1226:-tuple
1136:origin
1134:as an
897:and a
846:forces
762:vector
706:under
684:vector
677:origin
669:ABB′A′
574:couple
513:vector
495:and a
409:, and
356:vector
352:scalar
317:vector
265:forces
255:: the
236:, and
224:, and
189:vector
92:) and
90:length
74:vector
64:, and
13095:Minor
13080:Block
13018:Basis
12337:(PDF)
12330:(PDF)
12173:
12074:Notes
11827:force
11709:basis
11434:force
11387:Force
11280:speed
10814:scale
9862:or a
9801:from
9789:from
6530:seven
6372:(see
6362:angle
5598:and 2
5416:scale
5306:from
5098:and (
3732:) or
3667:basis
3665:as a
3574:above
3184:) of
2112:tuple
2098:in a
2066:arrow
2061:plane
2013:, or
1958:. In
1865:tilde
1707:field
1656:tuple
1619:In a
1577:angle
1298:force
1286:speed
1268:of 4
1242:force
1138:. By
943:(See
710:, of
671:is a
566:force
535:, or
507:. In
341:of a
196:from
13250:Dual
13105:Rank
12842:ISBN
12818:ISBN
12790:ISBN
12764:ISBN
12713:ISBN
12690:ISBN
12669:ISBN
12643:ISBN
12606:OCLC
12596:ISBN
12552:ISBN
12498:and
12486:and
12209:ISBN
12181:and
12144:veho
11744:′ =
11727:′ =
11703:. A
11579:any
11577:cite
11436:and
11271:The
10863:and
7131:and
7012:The
6750:and
6723:and
6672:and
6656:and
6640:and
6544:and
6516:The
6380:and
6368:and
6259:and
6251:The
6174:The
5690:norm
5675:The
5643:and
5625:) =
5369:and
5344:(-b)
5336:(-b)
5328:(-b)
5314:and
5129:and
5106:) +
5070:and
5058:and
5050:and
5035:and
4993:and
4816:and
4790:(or
4564:and
4237:and
3817:and
3180:(or
2128:(or
2023:head
2011:base
2007:tail
1792:and
1592:and
1590:area
1347:and
1320:and
1254:axis
939:and
937:free
848:and
726:and
616:and
572:and
548:and
511:, a
487:and
443:and
287:and
259:and
100:. A
88:(or
68:, a
12884:PDF
11994:'s
11904:car
11887:or
11865:and
11848:of
11649:to
11590:by
11264:to
11052:= (
11048:),
11027:= (
10883:= (
9841:".
9837:to
9829:to
9813:to
9805:to
9793:to
9248:to
7798:= {
7020:or
6684:).
6666:two
6600:sin
6524:or
6333:cos
6237:).
6219:is
5952:= (
5687:or
5681:or
5575:If
5133:is
5122:).
5114:+ (
4989:of
3852:).
3828:of
3588:or
3000:or
2817:in
2801:-,
2499:or
2385:).
2223:In
2037:or
2027:tip
1952:or
1904:or
1857:or
1820:. (
1735:to
1686:).
1666:or
1631:).
1402:= 1
1391:= 1
1094:is
1081:the
905:of
832:to
576:).
483:In
457:'s
271:or
208:on
200:to
80:or
56:In
49:to
13348::
12871:,
12865:,
12736:,
12730:,
12711:,
12604:.
12566:^
12534:^
12435:.
12424:^
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12320:,
12306:^
12290:.
12265:.
12250:^
12234:.
12223:^
12161:.
12156:n.
11922:,
11860:.
11833:.
11825:,
11821:,
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11813:,
11788:,
11548:.
11066:,
11059:,
11041:,
11034:,
11020:.
10897:,
10890:,
10783:A
9782:jk
9765:,
9758:,
9740:,
9733:,
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9624:32
9576:31
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9092:23
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8945:33
8929:32
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8883:23
8867:22
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8821:13
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8789:11
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8067:,
8063:,
8055:,
8051:,
7812:,
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7556:,
7549:,
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6735:.
6707:,
6396:.
6279:b,
6232:=
6228:+
6211:,
5966:,
5959:,
5944:.
5932:A
5835:,
5828:,
5722:,
5667:.
5659:=
5655:−
5632:+
5621:+
5358:.
5354:−
5350:=
5346:+
5330:+
5118:+
5110:=
5102:+
5094:+
5090:=
5086:+
5078:+
5062:+
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4997:.
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3373:,
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2009:,
2005:,
1990:.
1955:AB
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1232:.
844:,
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770:,
751:,
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732:,
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332:=
291:.
248:.
232:,
220:,
216:,
60:,
12923:e
12916:t
12909:v
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12882:(
12850:.
12826:.
12798:.
12772:.
12751:.
12741:.
12722:.
12699:.
12677:.
12651:.
12612:.
12560:.
12528:.
12501:b
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12477:r
12446:.
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12401:(
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12300:.
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12217:.
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12171:.
12158:"
11802:v
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11790:y
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11684:(
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11643:.
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11584:.
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11492:1
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11469:x
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11455:F
11450:=
11447:W
11416:a
11411:m
11408:=
11403:F
11372:a
11359:t
11355:0
11352:x
11338:,
11333:0
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11321:+
11316:v
11311:t
11308:=
11303:t
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11284:t
11276:v
11266:y
11262:x
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11254:y
11240:.
11235:3
11229:e
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11148:+
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11113:1
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11105:(
11102:=
11097:x
11087:y
11071:3
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11064:2
11061:y
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11025:x
11004:.
10999:3
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10977:+
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10950:+
10945:1
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10931:1
10927:x
10923:=
10918:x
10902:3
10899:x
10895:2
10892:x
10888:1
10885:x
10881:x
10853:t
10851:(
10849:r
10845:r
10841:t
10769:.
10764:2
10759:e
10752:1
10747:e
10742:)
10737:1
10733:b
10727:2
10723:a
10714:2
10710:b
10704:1
10700:a
10696:(
10693:=
10690:)
10685:2
10679:e
10671:2
10667:b
10663:+
10658:1
10652:e
10644:1
10640:b
10636:(
10630:)
10625:2
10619:e
10611:2
10607:a
10603:+
10598:1
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10584:1
10580:a
10576:(
10539:.
10534:4
10528:e
10522:)
10517:4
10513:b
10509:+
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10496:(
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10488:3
10482:e
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10471:3
10467:b
10463:+
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10454:a
10450:(
10447:+
10438:2
10432:e
10426:)
10421:2
10417:b
10413:+
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10404:a
10400:(
10397:+
10392:1
10386:e
10380:)
10375:1
10371:b
10367:+
10362:1
10358:a
10354:(
10347:=
10344:)
10339:4
10333:e
10325:4
10321:b
10317:+
10312:3
10306:e
10298:3
10294:b
10290:+
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10236:(
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10215:e
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10199:+
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10172:+
10167:2
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10153:2
10149:a
10145:+
10140:1
10134:e
10126:1
10122:a
10118:(
10094:,
10089:2
10083:e
10077:)
10072:2
10068:b
10064:+
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10055:a
10051:(
10048:+
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10037:e
10031:)
10026:1
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10018:+
10013:1
10009:a
10005:(
10002:=
9999:)
9994:2
9988:e
9980:2
9976:b
9972:+
9967:1
9961:e
9953:1
9949:b
9945:(
9942:+
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9839:e
9835:n
9831:n
9827:e
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9811:e
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9774:n
9770:3
9767:n
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9760:n
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9749:e
9745:3
9742:e
9738:2
9735:e
9731:1
9728:e
9706:3
9701:e
9691:3
9686:n
9681:=
9668:c
9658:2
9653:e
9643:3
9638:n
9633:=
9620:c
9610:1
9605:e
9595:3
9590:n
9585:=
9572:c
9562:3
9557:e
9547:2
9542:n
9537:=
9524:c
9514:2
9509:e
9499:2
9494:n
9489:=
9476:c
9466:1
9461:e
9451:2
9446:n
9441:=
9428:c
9418:3
9413:e
9403:1
9398:n
9393:=
9380:c
9370:2
9365:e
9355:1
9350:n
9345:=
9332:c
9322:1
9317:e
9307:1
9302:n
9297:=
9284:c
9254:k
9250:e
9245:j
9241:n
9228:c
9224:r
9220:q
9216:p
9212:e
9208:w
9204:v
9202:,
9200:u
9196:n
9192:a
9176:.
9171:]
9165:r
9158:q
9151:p
9145:[
9138:]
9126:c
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9102:c
9088:c
9076:c
9064:c
9050:c
9038:c
9026:c
9019:[
9014:=
9009:]
9003:w
8996:v
8989:u
8983:[
8953:,
8950:r
8941:c
8937:+
8934:q
8925:c
8921:+
8918:p
8909:c
8905:=
8898:w
8891:,
8888:r
8879:c
8875:+
8872:q
8863:c
8859:+
8856:p
8847:c
8843:=
8836:v
8829:,
8826:r
8817:c
8813:+
8810:q
8801:c
8797:+
8794:p
8785:c
8781:=
8774:u
8742:.
8737:3
8732:n
8722:3
8717:e
8712:r
8709:+
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8676:+
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8656:1
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8646:p
8643:=
8636:w
8629:,
8624:2
8619:n
8609:3
8604:e
8599:r
8596:+
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8576:2
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8563:+
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8553:n
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8533:p
8530:=
8523:v
8516:,
8511:1
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8496:3
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8463:2
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8450:+
8445:1
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8425:e
8420:p
8417:=
8410:u
8378:.
8373:3
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8360:)
8355:3
8350:e
8345:r
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8332:e
8327:q
8324:+
8319:1
8314:e
8309:p
8306:(
8303:=
8296:w
8289:,
8284:2
8279:n
8271:)
8266:3
8261:e
8256:r
8253:+
8248:2
8243:e
8238:q
8235:+
8230:1
8225:e
8220:p
8217:(
8214:=
8207:v
8200:,
8195:1
8190:n
8182:)
8177:3
8172:e
8167:r
8164:+
8159:2
8154:e
8149:q
8146:+
8141:1
8136:e
8131:p
8128:(
8125:=
8118:u
8098:r
8094:q
8090:p
8086:w
8082:v
8078:u
8073:a
8069:w
8065:v
8061:u
8057:r
8053:q
8049:p
8029:.
8024:3
8019:n
8010:a
8006:=
7999:w
7992:,
7987:2
7982:n
7973:a
7969:=
7962:v
7955:,
7950:1
7945:n
7936:a
7932:=
7925:u
7909:n
7891:3
7886:n
7881:w
7878:+
7873:2
7868:n
7863:v
7860:+
7855:1
7850:n
7845:u
7842:=
7838:a
7825:a
7821:e
7817:3
7814:n
7810:2
7807:n
7803:1
7800:n
7796:n
7776:.
7771:3
7766:e
7757:a
7753:=
7746:r
7739:,
7734:2
7729:e
7720:a
7716:=
7709:q
7702:,
7697:1
7692:e
7683:a
7679:=
7672:p
7656:e
7640:.
7635:3
7630:e
7625:r
7622:+
7617:2
7612:e
7607:q
7604:+
7599:1
7594:e
7589:p
7586:=
7582:a
7569:a
7565:e
7561:3
7558:e
7554:2
7551:e
7547:1
7544:e
7540:e
7519:.
7516:)
7512:a
7504:b
7496:c
7492:(
7486:=
7483:)
7479:c
7471:a
7463:b
7459:(
7453:=
7450:)
7446:b
7438:c
7430:a
7426:(
7420:=
7417:)
7413:a
7405:c
7397:b
7393:(
7390:=
7387:)
7383:b
7375:a
7367:c
7363:(
7360:=
7357:)
7353:c
7345:b
7337:a
7333:(
7309:|
7301:3
7297:c
7289:2
7285:c
7277:1
7273:c
7263:3
7259:b
7251:2
7247:b
7239:1
7235:b
7225:3
7221:a
7213:2
7209:a
7201:1
7197:a
7190:|
7185:=
7182:)
7178:c
7170:b
7162:a
7158:(
7133:c
7129:b
7125:a
7101:.
7098:)
7094:c
7086:b
7082:(
7075:a
7071:=
7068:)
7064:c
7056:b
7048:a
7044:(
7032:c
7029:b
7026:a
6980:.
6975:3
6969:e
6963:)
6958:1
6954:b
6948:2
6944:a
6935:2
6931:b
6925:1
6921:a
6917:(
6914:+
6909:2
6903:e
6897:)
6892:3
6888:b
6882:1
6878:a
6869:1
6865:b
6859:3
6855:a
6851:(
6848:+
6843:1
6837:e
6831:)
6826:2
6822:b
6816:3
6812:a
6803:3
6799:b
6793:2
6789:a
6785:(
6782:=
6777:b
6767:a
6752:b
6748:a
6744:b
6740:a
6725:b
6721:a
6717:b
6713:a
6709:b
6705:a
6701:b
6697:a
6682:n
6678:n
6674:b
6670:a
6658:b
6654:a
6646:n
6642:b
6638:a
6634:θ
6617:n
6612:)
6606:(
6592:b
6579:a
6571:=
6567:b
6559:a
6546:b
6542:a
6538:b
6534:a
6491:.
6486:3
6482:b
6476:3
6472:a
6468:+
6463:2
6459:b
6453:2
6449:a
6445:+
6440:1
6436:b
6430:1
6426:a
6422:=
6418:b
6410:a
6394:a
6390:b
6386:a
6382:b
6378:a
6370:b
6366:a
6358:θ
6342:,
6325:b
6312:a
6304:=
6300:b
6292:a
6275:a
6261:b
6257:a
6234:a
6230:a
6226:0
6221:a
6217:a
6213:0
6193:0
6147:3
6142:e
6130:a
6120:3
6116:a
6110:+
6105:2
6100:e
6088:a
6078:2
6074:a
6068:+
6063:1
6058:e
6046:a
6036:1
6032:a
6026:=
6016:a
6007:a
6001:=
5992:a
5977:a
5973:)
5971:3
5968:a
5964:2
5961:a
5957:1
5954:a
5950:a
5942:â
5921:â
5917:a
5892:.
5886:a
5878:a
5872:=
5864:a
5840:3
5837:e
5833:2
5830:e
5826:1
5823:e
5803:,
5796:2
5791:3
5787:a
5783:+
5778:2
5773:2
5769:a
5765:+
5760:2
5755:1
5751:a
5745:=
5737:a
5714:a
5703:a
5699:a
5695:a
5665:b
5661:a
5657:b
5653:a
5649:r
5645:b
5641:a
5637:b
5634:r
5630:a
5627:r
5623:b
5619:a
5617:(
5615:r
5604:a
5600:a
5596:a
5585:r
5581:r
5577:r
5570:r
5566:r
5562:r
5558:r
5542:.
5537:3
5532:e
5527:)
5522:3
5518:a
5514:r
5511:(
5508:+
5503:2
5498:e
5493:)
5488:2
5484:a
5480:r
5477:(
5474:+
5469:1
5464:e
5459:)
5454:1
5450:a
5446:r
5443:(
5440:=
5436:a
5432:r
5404:r
5371:b
5367:a
5356:b
5352:a
5348:a
5340:b
5332:a
5324:a
5320:b
5316:b
5312:a
5308:a
5304:b
5288:.
5283:3
5278:e
5273:)
5268:3
5264:b
5255:3
5251:a
5247:(
5244:+
5239:2
5234:e
5229:)
5224:2
5220:b
5211:2
5207:a
5203:(
5200:+
5195:1
5190:e
5185:)
5180:1
5176:b
5167:1
5163:a
5159:(
5156:=
5152:b
5144:a
5131:b
5127:a
5120:c
5116:b
5112:a
5108:c
5104:b
5100:a
5096:a
5092:b
5088:b
5084:a
5080:b
5076:a
5072:b
5068:a
5064:b
5060:a
5052:b
5048:a
5037:b
5033:a
5022:b
5018:a
5014:b
5010:a
5006:a
5002:b
4995:b
4991:a
4973:.
4968:3
4963:e
4958:)
4953:3
4949:b
4945:+
4940:3
4936:a
4932:(
4929:+
4924:2
4919:e
4914:)
4909:2
4905:b
4901:+
4896:2
4892:a
4888:(
4885:+
4880:1
4875:e
4870:)
4865:1
4861:b
4857:+
4852:1
4848:a
4844:(
4841:=
4837:b
4833:+
4829:a
4818:b
4814:a
4769:.
4764:3
4760:b
4753:=
4748:3
4744:a
4739:,
4734:2
4730:b
4723:=
4718:2
4714:a
4709:,
4704:1
4700:b
4693:=
4688:1
4684:a
4657:3
4651:e
4643:3
4639:b
4635:+
4630:2
4624:e
4616:2
4612:b
4608:+
4603:1
4597:e
4589:1
4585:b
4581:=
4576:b
4548:3
4542:e
4534:3
4530:a
4526:+
4521:2
4515:e
4507:2
4503:a
4499:+
4494:1
4488:e
4480:1
4476:a
4472:=
4467:a
4427:.
4422:3
4418:b
4414:=
4409:3
4405:a
4400:,
4395:2
4391:b
4387:=
4382:2
4378:a
4373:,
4368:1
4364:b
4360:=
4355:1
4351:a
4328:3
4322:e
4314:3
4310:b
4306:+
4301:2
4295:e
4287:2
4283:b
4279:+
4274:1
4268:e
4260:1
4256:b
4252:=
4247:b
4223:3
4217:e
4209:3
4205:a
4201:+
4196:2
4190:e
4182:2
4178:a
4174:+
4169:1
4163:e
4155:1
4151:a
4147:=
4142:a
4113:.
4108:3
4102:e
4094:3
4090:a
4086:+
4081:2
4075:e
4067:2
4063:a
4059:+
4054:1
4048:e
4040:1
4036:a
4032:=
4027:a
4015:a
4001:)
3998:1
3995:,
3992:0
3989:,
3986:0
3983:(
3980:=
3975:3
3969:e
3960:,
3957:)
3954:0
3951:,
3948:1
3945:,
3942:0
3939:(
3936:=
3931:2
3925:e
3916:,
3913:)
3910:0
3907:,
3904:0
3901:,
3898:1
3895:(
3892:=
3887:1
3881:e
3771:,
3756:,
3747:r
3736:(
3714:z
3708:,
3693:,
3673:(
3647:z
3641:,
3632:y
3626:,
3617:x
3547:i
3543:e
3527:.
3522:k
3515:z
3511:a
3507:+
3502:j
3495:y
3491:a
3487:+
3482:i
3475:x
3471:a
3467:=
3462:z
3457:a
3452:+
3447:y
3442:a
3437:+
3432:x
3427:a
3422:=
3418:a
3404:z
3400:a
3395:y
3391:a
3386:x
3382:a
3377:z
3375:a
3370:y
3368:a
3363:x
3361:a
3307:z
3301:,
3292:y
3286:,
3277:x
3253:k
3249:,
3245:j
3241:,
3237:i
3219:3
3216:a
3212:2
3209:a
3205:1
3202:a
3198:z
3194:y
3190:x
3186:a
3172:3
3169:a
3165:2
3162:a
3158:1
3155:a
3139:,
3134:3
3128:e
3120:3
3116:a
3112:+
3107:2
3101:e
3093:2
3089:a
3085:+
3080:1
3074:e
3066:1
3062:a
3058:=
3053:3
3048:a
3043:+
3038:2
3033:a
3028:+
3023:1
3018:a
3013:=
3009:a
2983:,
2980:)
2977:1
2974:,
2971:0
2968:,
2965:0
2962:(
2957:3
2953:a
2949:+
2946:)
2943:0
2940:,
2937:1
2934:,
2931:0
2928:(
2923:2
2919:a
2915:+
2912:)
2909:0
2906:,
2903:0
2900:,
2897:1
2894:(
2889:1
2885:a
2881:=
2878:)
2873:3
2869:a
2865:,
2860:2
2856:a
2852:,
2847:1
2843:a
2839:(
2836:=
2832:a
2820:R
2815:a
2807:z
2803:y
2799:x
2785:.
2782:)
2779:1
2776:,
2773:0
2770:,
2767:0
2764:(
2761:=
2756:3
2750:e
2741:,
2738:)
2735:0
2732:,
2729:1
2726:,
2723:0
2720:(
2717:=
2712:2
2706:e
2697:,
2694:)
2691:0
2688:,
2685:0
2682:,
2679:1
2676:(
2673:=
2668:1
2662:e
2645:n
2629:.
2624:T
2620:]
2614:3
2610:a
2601:2
2597:a
2588:1
2584:a
2580:[
2577:=
2572:]
2564:3
2560:a
2550:2
2546:a
2536:1
2532:a
2525:[
2520:=
2516:a
2481:.
2478:)
2473:n
2469:a
2465:,
2460:1
2454:n
2450:a
2446:,
2440:,
2435:3
2431:a
2427:,
2422:2
2418:a
2414:,
2409:1
2405:a
2401:(
2398:=
2394:a
2382:R
2361:.
2358:)
2353:z
2349:a
2345:,
2340:y
2336:a
2332:,
2327:x
2323:a
2319:(
2316:=
2312:a
2291:.
2288:)
2283:3
2279:a
2275:,
2270:2
2266:a
2262:,
2257:1
2253:a
2249:(
2246:=
2242:a
2230:R
2202:A
2199:O
2174:.
2171:)
2168:3
2165:,
2162:2
2159:(
2156:=
2152:a
2141:A
2137:O
2110:-
2108:n
2104:n
2092:n
2077:A
2019:B
1999:A
1976:a
1931:B
1928:A
1914:B
1910:A
1877:a
1860:a
1839:a
1817:a
1801:w
1778:v
1755:u
1737:B
1733:A
1680:K
1614:n
1606:n
1552:.
1548:)
1545:7
1542:,
1539:2
1536:,
1533:1
1527:(
1524:=
1521:)
1518:4
1515:+
1512:3
1509:,
1506:0
1503:+
1500:2
1497:,
1494:2
1488:1
1485:(
1482:=
1479:)
1476:4
1473:,
1470:0
1467:,
1464:2
1458:(
1455:+
1452:)
1449:3
1446:,
1443:2
1440:,
1437:1
1434:(
1420:x
1414:O
1406:y
1400:y
1395:x
1389:x
1368:B
1365:A
1350:B
1343:A
1294:y
1266:s
1262:F
1258:F
1246:F
1224:n
1209:,
1204:n
1199:R
1177:.
1168:P
1165:O
1152:P
1132:O
1118:.
1113:n
1108:R
1092:n
1088:n
1065:n
1060:R
1033:n
1028:R
1000:n
995:R
957:E
923:,
915:E
885:,
877:E
861:E
834:B
830:A
826:)
824:B
820:A
818:(
799:.
790:B
787:A
774:)
772:B
768:A
766:(
753:C
749:D
745:B
741:A
736:)
734:D
730:C
728:(
724:)
722:B
718:A
716:(
637:B
629:A
594:B
591:A
377:v
347:s
338:v
334:s
330:q
202:B
198:A
185:B
181:A
165:.
153:B
150:A
136:B
129:A
51:B
47:A
38:.
20:)
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