255:
2002:
polar vector. Angular momentum is the cross product of a displacement (a polar vector) and momentum (a polar vector), and is therefore a pseudovector. Torque is angular momentum (a pseudovector) divided by time (a scalar), so is also a pseudovector. Continuing this way, it is straightforward to classify any of the common vectors in physics as either a pseudovector or polar vector. (There are the parity-violating vectors in the theory of weak-interactions, which are neither polar vectors nor pseudovectors. However, these occur very rarely in physics.)
1536:
538:, which is a matrix representing a rank two mixed tensor, with one contravariant and one covariant index. As such, the noncommutativity of standard matrix algebra can be used to keep track of the distinction between covariant and contravariant vectors. This is in fact how the bookkeeping was done before the more formal and generalised tensor notation came to be. It still manifests itself in how the basis vectors of general tensor spaces are exhibited for practical manipulation.
390:, and pseudovectors are represented in this form too. When transforming between left and right-handed coordinate systems, representations of pseudovectors do not transform as vectors, and treating them as vector representations will cause an incorrect sign change, so that care must be taken to keep track of which ordered triplets represent vectors, and which represent pseudovectors. This problem does not exist if the cross product of two vectors is replaced by the
29:
1064:
1964:
3526:
3177:
726:
307:(invariant) under mirror reflections through this plane, with the magnetic field unchanged by the reflection. But reflecting the magnetic field as a vector through that plane would be expected to reverse it; this expectation is corrected by realizing that the magnetic field is a pseudovector, with the extra sign flip leaving it unchanged.
1506:
1324:
808:
281:. Driving in a car, and looking forward, each of the wheels has an angular momentum vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, the "reflection" of this angular momentum "vector" (viewed as an ordinary vector) points to the right, but the
1076:
is also a pseudovector. Similarly one can show that the difference between two pseudovectors is a pseudovector, that the sum or difference of two polar vectors is a polar vector, that multiplying a polar vector by any real number yields another polar vector, and that multiplying a pseudovector by any
318:
of a polar vector field. The cross product and curl are defined, by convention, according to the right hand rule, but could have been just as easily defined in terms of a left-hand rule. The entire body of physics that deals with (right-handed) pseudovectors and the right hand rule could be replaced
3727:
As an aside, it may be noted that not all authors in the field of geometric algebra use the term pseudovector, and some authors follow the terminology that does not distinguish between the pseudovector and the cross product. However, because the cross product does not generalize to other than three
2001:
From the definition, it is clear that a displacement vector is a polar vector. The velocity vector is a displacement vector (a polar vector) divided by time (a scalar), so is also a polar vector. Likewise, the momentum vector is the velocity vector (a polar vector) times mass (a scalar), so is a
3718:
are inverted, then the pseudovector is invariant, but the cross product changes sign. This behavior of cross products is consistent with their definition as vector-like elements that change sign under transformation from a right-handed to a left-handed coordinate system, unlike polar vectors.
258:
Each wheel of the car on the left driving away from an observer has an angular momentum pseudovector pointing left. The same is true for the mirror image of the car. The fact that the arrows point in the same direction, rather than being mirror images of each other indicates that they are
3291:
1751:
2945:
618:
1667:
3612:
533:
A basic and rather concrete example is that of row and column vectors under the usual matrix multiplication operator: in one order they yield the dot product, which is just a scalar and as such a rank zero tensor, while in the other they yield the
322:
While vector relationships in physics can be expressed in a coordinate-free manner, a coordinate system is required in order to express vectors and pseudovectors as numerical quantities. Vectors are represented as ordered triplets of numbers: e.g.
1059:{\displaystyle {\begin{aligned}\mathbf {v_{3}} '=\mathbf {v_{1}} '+\mathbf {v_{2}} '&=(\det R)(R\mathbf {v_{1}} )+(\det R)(R\mathbf {v_{2}} )\\&=(\det R)(R(\mathbf {v_{1}} +\mathbf {v_{2}} ))=(\det R)(R\mathbf {v_{3}} ).\end{aligned}}}
1349:
1141:
2030:. Any polar vector (e.g., a translation vector) would be unchanged, but pseudovectors (e.g., the magnetic field vector at a point) would switch signs. Nevertheless, there would be no physical consequences, apart from in the
3710:
are inverted by changing the signs of their components while leaving the basis vectors fixed, both the pseudovector and the cross product are invariant. On the other hand, if the components are fixed and the basis vectors
2732:
2560:
2423:
2265:
3521:{\displaystyle \mathbf {a} \wedge \mathbf {b} =\left(a^{2}b^{3}-a^{3}b^{2}\right)\mathbf {e} _{23}+\left(a^{3}b^{1}-a^{1}b^{3}\right)\mathbf {e} _{31}+\left(a^{1}b^{2}-a^{2}b^{1}\right)\mathbf {e} _{12}\ .}
525:
rank one. In this more general framework, higher rank tensors can also have arbitrarily many and mixed covariant and contravariant ranks at the same time, denoted by raised and lowered indices within the
428:: In particular, if everything in the universe were rotated, the vector would rotate in exactly the same way. (The coordinate system is fixed in this discussion; in other words this is the perspective of
1959:{\displaystyle \mathbf {v_{3}} '=\mathbf {v_{1}} '\times \mathbf {v_{2}} '=(R\mathbf {v_{1}} )\times (R\mathbf {v_{2}} )=(\det R)(R(\mathbf {v_{1}} \times \mathbf {v_{2}} ))=(\det R)(R\mathbf {v_{3}} ).}
3172:{\displaystyle \mathbf {a} \times \mathbf {b} =\left(a^{2}b^{3}-a^{3}b^{2}\right)\mathbf {e} _{1}+\left(a^{3}b^{1}-a^{1}b^{3}\right)\mathbf {e} _{2}+\left(a^{1}b^{2}-a^{2}b^{1}\right)\mathbf {e} _{3},}
1519:
were to describe a measurable physical quantity, that would mean that the laws of physics would not appear the same if the universe was viewed in a mirror. In fact, this is exactly what happens in the
2661:
the basic elements are vectors, and these are used to build a hierarchy of elements using the definitions of products in this algebra. In particular, the algebra builds pseudovectors from vectors.
813:
623:
2477:
398:
which is a 2nd rank tensor and is represented by a 3×3 matrix. This representation of the 2-tensor transforms correctly between any two coordinate systems, independently of their handedness.
388:
721:{\displaystyle {\begin{aligned}\mathbf {v} '&=R\mathbf {v} &&{\text{(polar vector)}}\\\mathbf {v} '&=(\det R)(R\mathbf {v} )&&{\text{(pseudovector)}}\end{aligned}}}
285:
angular momentum vector of the wheel (which is still turning forward in the reflection) still points to the left, corresponding to the extra sign flip in the reflection of a pseudovector.
416:
The definition of a "vector" in physics (including both polar vectors and pseudovectors) is more specific than the mathematical definition of "vector" (namely, any element of an abstract
1523:: Certain radioactive decays treat "left" and "right" differently, a phenomenon which can be traced to the summation of a polar vector with a pseudovector in the underlying theory. (See
1553:
2358:
319:
by using (left-handed) pseudovectors and the left hand rule without issue. The (left) pseudovectors so defined would be opposite in direction to those defined by the right-hand rule.
3697:
2587:
2895:
are often treated as synonymous, but it is quite useful to be able to distinguish a bivector from its dual." To paraphrase Baylis: Given two polar vectors (that is, true vectors)
2745:. Using the postulates of the algebra, all combinations of dot and wedge products can be evaluated. A terminology to describe the various combinations is provided. For example, a
3543:
2647:
2618:
2316:
2200:
2165:
2113:
is 2, a sign flip has no effect. Otherwise, the definitions are equivalent, though it should be borne in mind that without additional structure (specifically, either a
1501:{\displaystyle |\mathbf {v_{3}} |=|\mathbf {v_{1}} +\mathbf {v_{2}} |,{\text{ but }}\left|\mathbf {v_{3}} '\right|=\left|\mathbf {v_{1}} '-\mathbf {v_{2}} '\right|}
1319:{\displaystyle \mathbf {v_{3}} '=\mathbf {v_{1}} '+\mathbf {v_{2}} '=(R\mathbf {v_{1}} )+(\det R)(R\mathbf {v_{2}} )=R(\mathbf {v_{1}} +(\det R)\mathbf {v_{2}} ).}
3747:. The idea that "a pseudovector is different from a vector" is only true with a different and more specific definition of the term "vector" as discussed above.
3728:
dimensions, the notion of pseudovector based upon the cross product also cannot be extended to a space of any other number of dimensions. The pseudovector as a
2497:
2285:
3975:
2093:
This definition is not equivalent to that requiring a sign flip under improper rotations, but it is general to all vector spaces. In particular, when
4002:
46:(blue). If the position and current of the wire are reflected across the plane indicated by the dashed line, the magnetic field it generates would
2678:
2502:
2363:
3798:
2205:
1687:
are any three-dimensional vectors. (This equation can be proven either through a geometric argument or through an algebraic calculation.)
3893:
4500:
549:
in 3-dimensional space.) Suppose everything in the universe undergoes an improper rotation described by the improper rotation matrix
3646:
1336:
is neither a polar vector nor a pseudovector (although it is still a vector, by the physics definition). For an improper rotation,
522:
407:
4071:
506:
be considered the three components of a vector, since rotating the box does not appropriately transform these three components.)
2787:
can be expressed as the wedge product of two vectors and is a pseudovector. In four dimensions, however, the pseudovectors are
110:
is a pseudovector because it is often described as a vector, but by just changing the position of reference (and changing the
4337:
4207:
4142:
4113:
3985:
147:
will determine which), and is a pseudovector. This has consequences in computer graphics, where it has to be considered when
2428:
326:
4471:
4431:
4404:
4385:
4318:
4289:
4262:
4234:
4174:
4083:
4054:
4021:
3949:
3903:
3808:
2015:
2011:
541:
The discussion so far only relates to proper rotations, i.e. rotations about an axis. However, one can also consider
429:
3743:
Another important note is that pseudovectors, despite their name, are "vectors" in the sense of being elements of a
739:; this formula works because the determinant of proper and improper rotation matrices are +1 and −1, respectively.
79:
3182:
where superscripts label vector components. On the other hand, the plane of the two vectors is represented by the
1662:{\displaystyle (R\mathbf {v_{1}} )\times (R\mathbf {v_{2}} )=(\det R)(R(\mathbf {v_{1}} \times \mathbf {v_{2}} ))}
3849:
3824:
502:) from any other triplet of physical quantities (For example, the length, width, and height of a rectangular box
4373:
2321:
3656:
527:
2815:
basis pseudovectors. Each basis pseudovector is formed from the outer (wedge) product of all but one of the
114:), angular momentum can reverse direction, which is not supposed to happen with true vectors (also known as
2565:
545:, i.e. a mirror-reflection possibly followed by a proper rotation. (One example of an improper rotation is
3533:
4355:
3607:{\displaystyle \mathbf {a} \ \wedge \ \mathbf {b} ={\mathit {i}}\ \mathbf {a} \ \times \ \mathbf {b} \ ,}
2589:
is a direct product of group homomorphisms; it is the direct product of the fundamental homomorphism on
2883:
The transformation properties of the pseudovector in three dimensions has been compared to that of the
2168:
2118:
2623:
1993:
This is isomorphic to addition modulo 2, where "polar" corresponds to 1 and "pseudo" to 0.
546:
2592:
148:
4495:
2290:
2174:
2139:
2102:
2055:
99:
4279:
2775:
is one of these combinations. This term is attached to a different multivector depending upon the
2071:
4490:
4046:
425:
248:
166:
A number of quantities in physics behave as pseudovectors rather than polar vectors, including
95:
54:. The position and current at any point in the wire are "true" vectors, but the magnetic field
4415:
4224:
4041:
Theoretical methods in the physical sciences: an introduction to problem solving using Maple V
3933:
4252:
4191:
4162:
4130:
4101:
2133:
510:
4364:
2884:
2780:
2746:
2098:
2022:" with "left-hand rule" everywhere in math and physics, including in the definition of the
288:
The distinction between polar vectors and pseudovectors becomes important in understanding
254:
212:
87:
83:
4277:
3796:
180:, from which the transformation rules of pseudovectors can be derived. More generally, in
8:
2819:
basis vectors. For instance, in four dimensions where the basis vectors are taken to be {
2106:
440:
289:
103:
3934:"Clifford algebra derivation of the characteristic hypersurfaces of Maxwell's equations"
612:
The transformation rules for polar vectors and pseudovectors can be compactly stated as
4039:
2482:
2270:
2027:
315:
152:
4467:
4427:
4400:
4381:
4333:
4314:
4285:
4258:
4230:
4203:
4170:
4138:
4109:
4079:
4050:
4017:
3981:
3945:
3899:
3891:
3804:
2665:
2658:
2035:
542:
432:.) Mathematically, if everything in the universe undergoes a rotation described by a
185:
211:, both of which gain an extra sign-flip under improper rotations compared to a true
3761:
3756:
3183:
2762:
2742:
2031:
1539:
Under inversion the two vectors change sign, but their cross product is invariant .
1535:
1524:
1520:
411:
391:
264:
236:
232:
171:
122:
107:
33:
4347:
2019:
1547:, either proper or improper, the following mathematical equation is always true:
433:
144:
111:
4351:
3919:
3880:
4158:
4106:
Geometric
Algebra for Computer Science: An Object-Oriented Approach to Geometry
3778:
535:
240:
167:
40:
3881:
RP Feynman: §52-5 Polar and axial vectors, Feynman
Lectures in Physics, Vol. 1
742:
4484:
4466:, Chicago Lectures in Physics, The University of Chicago Press, p. 126,
3965:
3772:
2023:
311:
160:
4072:"Application of conformal geometric algebra in computer vision and graphics"
3744:
2926:
417:
207:
201:
156:
3969:
2788:
2738:
2727:{\displaystyle \mathbf {ab} =\mathbf {a\cdot b} +\mathbf {a\wedge b} \ ,}
2114:
736:
67:
22:
4167:
New foundations for classical mechanics: Fundamental
Theories of Physics
2783:
vectors in the space). In three dimensions, the most general 2-blade or
2555:{\displaystyle {\text{O}}(n)\cong {\text{SO}}(n)\times \mathbb {Z} _{2}}
143:
is a normal to the plane (there are two normals, one on each side – the
4278:
Stephen A. Fulling; Michael N. Sinyakov; Sergei V. Tischchenko (2000).
3797:
Stephen A. Fulling; Michael N. Sinyakov; Sergei V. Tischchenko (2000).
3766:
3202:
174:. In mathematics, in three dimensions, pseudovectors are equivalent to
4222:
3898:(Reprint of 1968 Prentice-Hall ed.). Courier Dover. p. 125.
2418:{\displaystyle (\mathbb {R} ^{n},\rho _{\text{pseudo}},{\text{O}}(n))}
2132:
Another way to formalize them is by considering them as elements of a
299:
plane that inside the loop generates a magnetic field oriented in the
2776:
304:
244:
4099:
2260:{\displaystyle (\mathbb {R} ^{n},\rho _{\text{fund}},{\text{O}}(n))}
2101:, such a pseudovector does not experience a sign flip, and when the
3197:
2784:
499:
395:
176:
91:
28:
4189:
3973:
731:
where the symbols are as described above, and the rotation matrix
310:
In physics, pseudovectors are generally the result of taking the
63:
3892:
Aleksandr
Ivanovich Borisenko; Ivan Evgenʹevich Tarapov (1979).
2939:, the cross product is expressed in terms of its components as:
2360:. Pseudovectors transform in a pseudofundamental representation
1124:. If the universe is transformed by an improper rotation matrix
420:). Under the physics definition, a "vector" is required to have
125:. An oriented plane can be defined by two non-parallel vectors,
518:
228:
216:
188:, pseudovectors are the elements of the algebra with dimension
4226:
Geometric algebra with applications in science and engineering
4069:
421:
743:
Behavior under addition, subtraction, scalar multiplication
121:
One example of a pseudovector is the normal to an oriented
4420:
Lectures on
Clifford (geometric) algebras and applications
4014:
Lectures on
Clifford (geometric) algebras and applications
3702:
Using the above relations, it is seen that if the vectors
290:
the effect of symmetry on the solution to physical systems
16:
Physical quantity that changes sign with improper rotation
2664:
The basic multiplication in the geometric algebra is the
735:
can be either proper or improper. The symbol det denotes
4076:
Computer algebra and geometric algebra with applications
3931:
3769:, a generalization of pseudovector in Clifford algebra
3537:. The cross product and wedge product are related by:
2878:
1734:. If the universe is transformed by a rotation matrix
791:. If the universe is transformed by a rotation matrix
4102:"Figure 3.5: Duality of vectors and bivectors in 3-D"
3659:
3546:
3294:
2948:
2903:
in three dimensions, the cross product composed from
2681:
2626:
2595:
2568:
2505:
2485:
2431:
2366:
2324:
2293:
2273:
2208:
2177:
2142:
2046:
One way to formalize pseudovectors is as follows: if
1754:
1556:
1352:
1144:
811:
621:
593:. If it is a pseudovector, it will be transformed to
482:. This important requirement is what distinguishes a
329:
4254:
Multivectors and
Clifford algebra in electrodynamics
199:. The label "pseudo-" can be further generalized to
4281:Linearity and the mathematics of several variables
4135:Geometric Algebra with Applications in Engineering
4038:
3920:Feynman Lectures, 52-7, "Parity is not conserved!"
3850:"Details for IEV number 102-03-34: "polar vector""
3825:"Details for IEV number 102-03-33: "axial vector""
3800:Linearity and the mathematics of several variables
3740:-dimensional space is not restricted in this way.
3691:
3606:
3520:
3171:
2726:
2668:, denoted by simply juxtaposing two vectors as in
2641:
2612:
2581:
2554:
2491:
2471:
2417:
2352:
2310:
2279:
2259:
2194:
2159:
1958:
1661:
1500:
1343:does not in general even keep the same magnitude:
1318:
1058:
720:
382:
4416:"4. Applications of Clifford algebras in physics"
4250:
4223:Eduardo Bayro Corrochano; Garret Sobczyk (2001).
4128:
4100:Leo Dorst; Daniel Fontijne; Stephen Mann (2007).
3876:
3874:
3872:
3870:
2472:{\displaystyle \rho _{\text{pseudo}}(R)=\det(R)R}
2014:. An alternate approach, more along the lines of
4482:
4418:. In Abłamowicz, Rafał; Sobczyk, Garret (eds.).
4036:
4000:
2454:
1923:
1866:
1608:
1530:
1289:
1232:
1019:
962:
919:
883:
683:
383:{\displaystyle \mathbf {a} =(a_{x},a_{y},a_{z})}
3775:— discussion about non-orientable spaces.
2737:where the leading term is the customary vector
2086:form a vector space with the same dimension as
2010:Above, pseudovectors have been discussed using
513:, this requirement is equivalent to defining a
4157:
3867:
2803:is the dimension of the space and algebra. An
486:(which might be composed of, for example, the
4196:Geometric algebra and applications to physics
4190:Venzo De Sabbata; Bidyut Kumar Datta (2007).
4108:(2nd ed.). Morgan Kaufmann. p. 82.
4070:R Wareham, J Cameron & J Lasenby (2005).
3977:Geometric algebra and applications to physics
3974:Venzo De Sabbata; Bidyut Kumar Datta (2007).
3196:. In this context of geometric algebra, this
2911:is the vector normal to their plane given by
2761:-fold wedge product also is referred to as a
2018:, is to keep the universe fixed, but switch "
577:is a polar vector, it will be transformed to
50:be reflected: Instead, it would be reflected
4327:
4271:
3895:Vector and tensor analysis with applications
2479:. Another way to view this homomorphism for
1976:is a pseudovector. Similarly, one can show:
4308:
3534:Hodge star operator § Three dimensions
2121:), there is no natural identification of ⋀(
292:. Consider an electric current loop in the
227:Physical examples of pseudovectors include
4309:Arfken, George B.; Weber, Hans J. (2001).
3938:Deformations of mathematical structures II
3790:
2925:. Given a set of right-handed orthonormal
1989:pseudovector × polar vector = polar vector
1986:polar vector × pseudovector = polar vector
1983:pseudovector × pseudovector = pseudovector
1980:polar vector × polar vector = pseudovector
424:that "transform" in a certain way under a
4461:
4243:
3854:International Electrotechnical Vocabulary
3829:International Electrotechnical Vocabulary
2847:}, the pseudovectors can be written as: {
2629:
2542:
2372:
2214:
1077:real number yields another pseudovector.
4151:
3958:
3925:
2353:{\displaystyle \rho _{\text{fund}}(R)=R}
1534:
408:Covariance and contravariance of vectors
253:
163:of two polar vectors are pseudovectors.
27:
4372:
4328:Doran, Chris; Lasenby, Anthony (2007).
4216:
4122:
3994:
3842:
3817:
3692:{\displaystyle {\mathit {i}}^{2}=-1\ .}
4483:
4413:
4169:(2nd ed.). Springer. p. 60.
4030:
3885:
2005:
1711:is defined to be their cross product,
4192:"The pseudoscalar and imaginary unit"
4183:
4003:"§4.2.3 Higher-grade multivectors in
3257:, and so forth. That is, the dual of
3200:is called a pseudovector, and is the
2779:of the space (that is, the number of
2582:{\displaystyle \rho _{\text{pseudo}}}
102:, etc. This can also happen when the
2652:
222:
86:does not conform when the object is
78:) is a quantity that behaves like a
4394:
4346:
4311:Mathematical Methods for Physicists
4090:In three dimensions, a dual may be
2879:Transformations in three dimensions
1094:is known to be a pseudovector, and
394:of the two vectors, which yields a
32:A loop of wire (black), carrying a
13:
4063:
3663:
3571:
2741:and the second term is called the
133:, that span the plane. The vector
14:
4512:
4501:Vectors (mathematics and physics)
4284:. World Scientific. p. 340.
3803:. World Scientific. p. 343.
3271:, namely the subspace spanned by
3264:is the subspace perpendicular to
2743:wedge product or exterior product
2620:with the trivial homomorphism on
466:must be similarly transformed to
4438:: The dual of the wedge product
4352:"§52-5: Polar and axial vectors"
4330:Geometric Algebra for Physicists
4257:. World Scientific. p. 11.
3932:William M Pezzaglia Jr. (1992).
3722:
3594:
3580:
3562:
3548:
3502:
3434:
3366:
3304:
3296:
3156:
3088:
3020:
2958:
2950:
2753:-fold wedge products of various
2714:
2708:
2700:
2694:
2686:
2683:
2672:. This product is expressed as:
2642:{\displaystyle \mathbb {Z} _{2}}
2041:
1944:
1940:
1905:
1901:
1890:
1886:
1851:
1847:
1827:
1823:
1802:
1798:
1782:
1778:
1762:
1758:
1647:
1643:
1632:
1628:
1593:
1589:
1569:
1565:
1483:
1479:
1463:
1459:
1434:
1430:
1404:
1400:
1389:
1385:
1364:
1360:
1304:
1300:
1277:
1273:
1253:
1249:
1217:
1213:
1192:
1188:
1172:
1168:
1152:
1148:
1040:
1036:
1001:
997:
986:
982:
940:
936:
904:
900:
863:
859:
843:
839:
823:
819:
699:
665:
647:
628:
331:
4368:at Encyclopaedia of Mathematics
3936:. In Julian Ławrynowicz (ed.).
2887:by Baylis. He says: "The terms
1087:is known to be a polar vector,
4332:. Cambridge University Press.
3964:In four dimensions, such as a
3912:
2613:{\displaystyle {\text{SO}}(n)}
2607:
2601:
2534:
2528:
2517:
2511:
2463:
2457:
2448:
2442:
2412:
2409:
2403:
2367:
2341:
2335:
2305:
2299:
2254:
2251:
2245:
2209:
2189:
2183:
2154:
2148:
1950:
1932:
1929:
1920:
1914:
1911:
1881:
1875:
1872:
1863:
1857:
1839:
1833:
1815:
1656:
1653:
1623:
1617:
1614:
1605:
1599:
1581:
1575:
1557:
1411:
1379:
1371:
1354:
1310:
1295:
1286:
1268:
1259:
1241:
1238:
1229:
1223:
1205:
1046:
1028:
1025:
1016:
1010:
1007:
977:
971:
968:
959:
946:
928:
925:
916:
910:
892:
889:
880:
703:
692:
689:
680:
377:
338:
1:
4301:
3186:or wedge product, denoted by
2311:{\displaystyle {\text{O}}(n)}
2195:{\displaystyle {\text{O}}(n)}
2160:{\displaystyle {\text{O}}(n)}
1704:are known polar vectors, and
1531:Behavior under cross products
761:are known pseudovectors, and
528:Einstein summation convention
106:is changed. For example, the
4360:. Vol. 1. p. 52–6.
1101:is defined to be their sum,
768:is defined to be their sum,
553:, so that a position vector
314:of two polar vectors or the
149:transforming surface normals
82:in many situations, but its
7:
4462:Weinreich, Gabriel (1998),
4357:Feynman Lectures on Physics
4016:. Birkhäuser. p. 100.
3750:
3285:. With this understanding,
2771:In the present context the
2167:. Vectors transform in the
1996:
1080:On the other hand, suppose
151:. In three dimensions, the
10:
4517:
4422:. Birkhäuser. p. 100
4414:Baylis, William E (2004).
4397:Mathematics for Physicists
4251:Bernard Jancewicz (1988).
4163:"The vector cross product"
4129:Christian Perwass (2009).
3206:of the cross product. The
2169:fundamental representation
2034:phenomena such as certain
405:
401:
303:direction. This system is
263:Consider the pseudovector
20:
4378:Classical Electrodynamics
4229:. Springer. p. 126.
4078:. Springer. p. 330.
4037:William E Baylis (1994).
4001:William E Baylis (2004).
3980:. CRC Press. p. 64.
2499:odd is that in this case
2267:, so that for any matrix
547:inversion through a point
4198:. CRC Press. p. 53
4137:. Springer. p. 17.
4131:"§1.5.2 General vectors"
3968:, the pseudovectors are
3940:. Springer. p. 131
3784:
2082:). The pseudovectors of
104:orientation of the space
21:Not to be confused with
3650:. It has the property:
2811:basis vectors and also
2807:-dimensional space has
2016:passive transformations
4395:Lea, Susan M. (2004).
4045:. Birkhäuser. p.
3693:
3608:
3522:
3173:
2791:. In general, it is a
2728:
2643:
2614:
2583:
2556:
2493:
2473:
2419:
2354:
2312:
2281:
2261:
2196:
2161:
2066:is an element of the (
2012:active transformations
1960:
1663:
1543:For a rotation matrix
1540:
1502:
1320:
1060:
722:
430:active transformations
384:
260:
249:magnetic dipole moment
59:
4448:is the cross product
3694:
3609:
3523:
3174:
2729:
2644:
2615:
2584:
2557:
2494:
2474:
2420:
2355:
2313:
2282:
2262:
2197:
2162:
2058:vector space, then a
1961:
1664:
1538:
1503:
1321:
1061:
723:
511:differential geometry
385:
257:
31:
3657:
3544:
3292:
2946:
2885:vector cross product
2781:linearly independent
2679:
2624:
2593:
2566:
2503:
2483:
2429:
2364:
2322:
2291:
2271:
2206:
2175:
2140:
2134:representation space
1752:
1554:
1512:If the magnitude of
1350:
1142:
809:
619:
509:(In the language of
462:, then any "vector"
327:
4464:Geometrical Vectors
2202:with data given by
2070: − 1)-th
2006:The right-hand rule
441:displacement vector
159:at a point and the
88:rigidly transformed
3689:
3604:
3518:
3169:
2749:is a summation of
2724:
2639:
2610:
2579:
2552:
2489:
2469:
2415:
2350:
2308:
2277:
2257:
2192:
2157:
2105:of the underlying
2036:radioactive decays
1956:
1745:is transformed to
1659:
1541:
1498:
1316:
1135:is transformed to
1056:
1054:
802:is transformed to
718:
716:
557:is transformed to
543:improper rotations
446:is transformed to
380:
261:
60:
58:is a pseudovector.
4339:978-0-521-71595-9
4209:978-1-58488-772-0
4144:978-3-540-89067-6
4115:978-0-12-374942-0
3987:978-1-58488-772-0
3685:
3647:unit pseudoscalar
3600:
3592:
3586:
3578:
3560:
3554:
3531:For details, see
3514:
3217:is introduced as
2720:
2666:geometric product
2659:geometric algebra
2653:Geometric algebra
2599:
2576:
2526:
2509:
2492:{\displaystyle n}
2439:
2401:
2392:
2332:
2297:
2280:{\displaystyle R}
2243:
2234:
2181:
2146:
1421:
712:
657:
223:Physical examples
186:geometric algebra
4508:
4476:
4457:
4447:
4437:
4410:
4391:
4361:
4348:Feynman, Richard
4343:
4324:
4296:
4295:
4275:
4269:
4268:
4247:
4241:
4240:
4220:
4214:
4213:
4187:
4181:
4180:
4155:
4149:
4148:
4126:
4120:
4119:
4089:
4067:
4061:
4060:
4049:, see footnote.
4044:
4034:
4028:
4027:
3998:
3992:
3991:
3962:
3956:
3955:
3929:
3923:
3916:
3910:
3909:
3889:
3883:
3878:
3865:
3864:
3862:
3861:
3846:
3840:
3839:
3837:
3836:
3821:
3815:
3814:
3794:
3762:Clifford algebra
3757:Exterior algebra
3735:
3698:
3696:
3695:
3690:
3683:
3673:
3672:
3667:
3666:
3643:
3613:
3611:
3610:
3605:
3598:
3597:
3590:
3584:
3583:
3576:
3575:
3574:
3565:
3558:
3552:
3551:
3527:
3525:
3524:
3519:
3512:
3511:
3510:
3505:
3499:
3495:
3494:
3493:
3484:
3483:
3471:
3470:
3461:
3460:
3443:
3442:
3437:
3431:
3427:
3426:
3425:
3416:
3415:
3403:
3402:
3393:
3392:
3375:
3374:
3369:
3363:
3359:
3358:
3357:
3348:
3347:
3335:
3334:
3325:
3324:
3307:
3299:
3256:
3241:
3226:
3195:
3184:exterior product
3178:
3176:
3175:
3170:
3165:
3164:
3159:
3153:
3149:
3148:
3147:
3138:
3137:
3125:
3124:
3115:
3114:
3097:
3096:
3091:
3085:
3081:
3080:
3079:
3070:
3069:
3057:
3056:
3047:
3046:
3029:
3028:
3023:
3017:
3013:
3012:
3011:
3002:
3001:
2989:
2988:
2979:
2978:
2961:
2953:
2938:
2924:
2798:
2733:
2731:
2730:
2725:
2718:
2717:
2703:
2689:
2648:
2646:
2645:
2640:
2638:
2637:
2632:
2619:
2617:
2616:
2611:
2600:
2597:
2588:
2586:
2585:
2580:
2578:
2577:
2574:
2561:
2559:
2558:
2553:
2551:
2550:
2545:
2527:
2524:
2510:
2507:
2498:
2496:
2495:
2490:
2478:
2476:
2475:
2470:
2441:
2440:
2437:
2424:
2422:
2421:
2416:
2402:
2399:
2394:
2393:
2390:
2381:
2380:
2375:
2359:
2357:
2356:
2351:
2334:
2333:
2330:
2317:
2315:
2314:
2309:
2298:
2295:
2286:
2284:
2283:
2278:
2266:
2264:
2263:
2258:
2244:
2241:
2236:
2235:
2232:
2223:
2222:
2217:
2201:
2199:
2198:
2193:
2182:
2179:
2166:
2164:
2163:
2158:
2147:
2144:
2032:parity-violating
1965:
1963:
1962:
1957:
1949:
1948:
1947:
1910:
1909:
1908:
1895:
1894:
1893:
1856:
1855:
1854:
1832:
1831:
1830:
1811:
1807:
1806:
1805:
1791:
1787:
1786:
1785:
1771:
1767:
1766:
1765:
1733:
1668:
1666:
1665:
1660:
1652:
1651:
1650:
1637:
1636:
1635:
1598:
1597:
1596:
1574:
1573:
1572:
1525:parity violation
1521:weak interaction
1507:
1505:
1504:
1499:
1497:
1493:
1492:
1488:
1487:
1486:
1472:
1468:
1467:
1466:
1447:
1443:
1439:
1438:
1437:
1422:
1419:
1414:
1409:
1408:
1407:
1394:
1393:
1392:
1382:
1374:
1369:
1368:
1367:
1357:
1325:
1323:
1322:
1317:
1309:
1308:
1307:
1282:
1281:
1280:
1258:
1257:
1256:
1222:
1221:
1220:
1201:
1197:
1196:
1195:
1181:
1177:
1176:
1175:
1161:
1157:
1156:
1155:
1123:
1065:
1063:
1062:
1057:
1055:
1045:
1044:
1043:
1006:
1005:
1004:
991:
990:
989:
952:
945:
944:
943:
909:
908:
907:
872:
868:
867:
866:
852:
848:
847:
846:
832:
828:
827:
826:
790:
727:
725:
724:
719:
717:
713:
710:
707:
702:
672:
668:
658:
655:
652:
650:
635:
631:
608:
600:
592:
584:
573:. If the vector
572:
564:
481:
473:
461:
453:
412:Euclidean vector
392:exterior product
389:
387:
386:
381:
376:
375:
363:
362:
350:
349:
334:
298:
280:
265:angular momentum
237:angular momentum
233:angular velocity
194:
172:angular velocity
142:
108:angular momentum
4516:
4515:
4511:
4510:
4509:
4507:
4506:
4505:
4496:Vector calculus
4481:
4480:
4479:
4474:
4449:
4439:
4434:
4407:
4388:
4340:
4321:
4304:
4299:
4292:
4276:
4272:
4265:
4248:
4244:
4237:
4221:
4217:
4210:
4188:
4184:
4177:
4156:
4152:
4145:
4127:
4123:
4116:
4086:
4068:
4064:
4057:
4035:
4031:
4024:
4009:
3999:
3995:
3988:
3963:
3959:
3952:
3930:
3926:
3917:
3913:
3906:
3890:
3886:
3879:
3868:
3859:
3857:
3848:
3847:
3843:
3834:
3832:
3823:
3822:
3818:
3811:
3795:
3791:
3787:
3753:
3729:
3725:
3717:
3668:
3662:
3661:
3660:
3658:
3655:
3654:
3642:
3635:
3628:
3618:
3593:
3579:
3570:
3569:
3561:
3547:
3545:
3542:
3541:
3506:
3501:
3500:
3489:
3485:
3479:
3475:
3466:
3462:
3456:
3452:
3451:
3447:
3438:
3433:
3432:
3421:
3417:
3411:
3407:
3398:
3394:
3388:
3384:
3383:
3379:
3370:
3365:
3364:
3353:
3349:
3343:
3339:
3330:
3326:
3320:
3316:
3315:
3311:
3303:
3295:
3293:
3290:
3289:
3284:
3277:
3270:
3263:
3255:
3248:
3242:
3239:
3233:
3227:
3224:
3218:
3216:
3187:
3160:
3155:
3154:
3143:
3139:
3133:
3129:
3120:
3116:
3110:
3106:
3105:
3101:
3092:
3087:
3086:
3075:
3071:
3065:
3061:
3052:
3048:
3042:
3038:
3037:
3033:
3024:
3019:
3018:
3007:
3003:
2997:
2993:
2984:
2980:
2974:
2970:
2969:
2965:
2957:
2949:
2947:
2944:
2943:
2936:
2929:
2912:
2881:
2874:
2867:
2860:
2853:
2846:
2839:
2832:
2825:
2792:
2707:
2693:
2682:
2680:
2677:
2676:
2655:
2633:
2628:
2627:
2625:
2622:
2621:
2596:
2594:
2591:
2590:
2573:
2569:
2567:
2564:
2563:
2546:
2541:
2540:
2523:
2506:
2504:
2501:
2500:
2484:
2481:
2480:
2436:
2432:
2430:
2427:
2426:
2398:
2389:
2385:
2376:
2371:
2370:
2365:
2362:
2361:
2329:
2325:
2323:
2320:
2319:
2294:
2292:
2289:
2288:
2272:
2269:
2268:
2240:
2231:
2227:
2218:
2213:
2212:
2207:
2204:
2203:
2178:
2176:
2173:
2172:
2143:
2141:
2138:
2137:
2044:
2020:right-hand rule
2008:
1999:
1975:
1943:
1939:
1938:
1904:
1900:
1899:
1889:
1885:
1884:
1850:
1846:
1845:
1826:
1822:
1821:
1801:
1797:
1796:
1795:
1781:
1777:
1776:
1775:
1761:
1757:
1756:
1755:
1753:
1750:
1749:
1744:
1732:
1725:
1718:
1712:
1710:
1703:
1696:
1686:
1679:
1646:
1642:
1641:
1631:
1627:
1626:
1592:
1588:
1587:
1568:
1564:
1563:
1555:
1552:
1551:
1533:
1518:
1482:
1478:
1477:
1476:
1462:
1458:
1457:
1456:
1455:
1451:
1433:
1429:
1428:
1427:
1423:
1420: but
1418:
1410:
1403:
1399:
1398:
1388:
1384:
1383:
1378:
1370:
1363:
1359:
1358:
1353:
1351:
1348:
1347:
1342:
1335:
1303:
1299:
1298:
1276:
1272:
1271:
1252:
1248:
1247:
1216:
1212:
1211:
1191:
1187:
1186:
1185:
1171:
1167:
1166:
1165:
1151:
1147:
1146:
1145:
1143:
1140:
1139:
1134:
1122:
1115:
1108:
1102:
1100:
1093:
1086:
1075:
1053:
1052:
1039:
1035:
1034:
1000:
996:
995:
985:
981:
980:
950:
949:
939:
935:
934:
903:
899:
898:
873:
862:
858:
857:
856:
842:
838:
837:
836:
822:
818:
817:
816:
812:
810:
807:
806:
801:
789:
782:
775:
769:
767:
760:
753:
745:
715:
714:
709:
706:
698:
673:
664:
663:
660:
659:
654:
651:
646:
636:
627:
626:
622:
620:
617:
616:
598:
594:
582:
578:
562:
558:
498:-components of
471:
467:
451:
447:
434:rotation matrix
426:proper rotation
414:
404:
371:
367:
358:
354:
345:
341:
330:
328:
325:
324:
293:
267:
225:
189:
145:right-hand rule
134:
112:position vector
26:
17:
12:
11:
5:
4514:
4504:
4503:
4498:
4493:
4491:Linear algebra
4478:
4477:
4472:
4459:
4432:
4411:
4405:
4392:
4386:
4374:Jackson, J. D.
4370:
4362:
4344:
4338:
4325:
4319:
4305:
4303:
4300:
4298:
4297:
4290:
4270:
4263:
4242:
4235:
4215:
4208:
4182:
4175:
4159:David Hestenes
4150:
4143:
4121:
4114:
4084:
4062:
4055:
4029:
4022:
4007:
3993:
3986:
3957:
3950:
3924:
3911:
3904:
3884:
3866:
3841:
3816:
3809:
3788:
3786:
3783:
3782:
3781:
3779:Tensor density
3776:
3770:
3764:
3759:
3752:
3749:
3724:
3721:
3715:
3700:
3699:
3688:
3682:
3679:
3676:
3671:
3665:
3644:is called the
3640:
3633:
3626:
3615:
3614:
3603:
3596:
3589:
3582:
3573:
3568:
3564:
3557:
3550:
3529:
3528:
3517:
3509:
3504:
3498:
3492:
3488:
3482:
3478:
3474:
3469:
3465:
3459:
3455:
3450:
3446:
3441:
3436:
3430:
3424:
3420:
3414:
3410:
3406:
3401:
3397:
3391:
3387:
3382:
3378:
3373:
3368:
3362:
3356:
3352:
3346:
3342:
3338:
3333:
3329:
3323:
3319:
3314:
3310:
3306:
3302:
3298:
3282:
3275:
3268:
3261:
3253:
3246:
3237:
3231:
3222:
3214:
3180:
3179:
3168:
3163:
3158:
3152:
3146:
3142:
3136:
3132:
3128:
3123:
3119:
3113:
3109:
3104:
3100:
3095:
3090:
3084:
3078:
3074:
3068:
3064:
3060:
3055:
3051:
3045:
3041:
3036:
3032:
3027:
3022:
3016:
3010:
3006:
3000:
2996:
2992:
2987:
2983:
2977:
2973:
2968:
2964:
2960:
2956:
2952:
2934:
2880:
2877:
2872:
2865:
2858:
2851:
2844:
2837:
2830:
2823:
2799:-blade, where
2735:
2734:
2723:
2716:
2713:
2710:
2706:
2702:
2699:
2696:
2692:
2688:
2685:
2654:
2651:
2636:
2631:
2609:
2606:
2603:
2572:
2549:
2544:
2539:
2536:
2533:
2530:
2522:
2519:
2516:
2513:
2488:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2435:
2414:
2411:
2408:
2405:
2397:
2388:
2384:
2379:
2374:
2369:
2349:
2346:
2343:
2340:
2337:
2328:
2307:
2304:
2301:
2276:
2256:
2253:
2250:
2247:
2239:
2230:
2226:
2221:
2216:
2211:
2191:
2188:
2185:
2156:
2153:
2150:
2103:characteristic
2072:exterior power
2043:
2040:
2007:
2004:
1998:
1995:
1991:
1990:
1987:
1984:
1981:
1973:
1967:
1966:
1955:
1952:
1946:
1942:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1907:
1903:
1898:
1892:
1888:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1853:
1849:
1844:
1841:
1838:
1835:
1829:
1825:
1820:
1817:
1814:
1810:
1804:
1800:
1794:
1790:
1784:
1780:
1774:
1770:
1764:
1760:
1742:
1730:
1723:
1716:
1708:
1701:
1694:
1684:
1677:
1671:
1670:
1658:
1655:
1649:
1645:
1640:
1634:
1630:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1595:
1591:
1586:
1583:
1580:
1577:
1571:
1567:
1562:
1559:
1532:
1529:
1516:
1510:
1509:
1496:
1491:
1485:
1481:
1475:
1471:
1465:
1461:
1454:
1450:
1446:
1442:
1436:
1432:
1426:
1417:
1413:
1406:
1402:
1397:
1391:
1387:
1381:
1377:
1373:
1366:
1362:
1356:
1340:
1333:
1327:
1326:
1315:
1312:
1306:
1302:
1297:
1294:
1291:
1288:
1285:
1279:
1275:
1270:
1267:
1264:
1261:
1255:
1251:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1219:
1215:
1210:
1207:
1204:
1200:
1194:
1190:
1184:
1180:
1174:
1170:
1164:
1160:
1154:
1150:
1132:
1120:
1113:
1106:
1098:
1091:
1084:
1073:
1067:
1066:
1051:
1048:
1042:
1038:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1003:
999:
994:
988:
984:
979:
976:
973:
970:
967:
964:
961:
958:
955:
953:
951:
948:
942:
938:
933:
930:
927:
924:
921:
918:
915:
912:
906:
902:
897:
894:
891:
888:
885:
882:
879:
876:
874:
871:
865:
861:
855:
851:
845:
841:
835:
831:
825:
821:
815:
814:
799:
787:
780:
773:
765:
758:
751:
744:
741:
729:
728:
711:(pseudovector)
708:
705:
701:
697:
694:
691:
688:
685:
682:
679:
676:
674:
671:
667:
662:
661:
656:(polar vector)
653:
649:
645:
642:
639:
637:
634:
630:
625:
624:
536:dyadic product
403:
400:
379:
374:
370:
366:
361:
357:
353:
348:
344:
340:
337:
333:
259:pseudovectors.
241:magnetic field
224:
221:
168:magnetic field
41:magnetic field
15:
9:
6:
4:
3:
2:
4513:
4502:
4499:
4497:
4494:
4492:
4489:
4488:
4486:
4475:
4473:9780226890487
4469:
4465:
4460:
4456:
4452:
4446:
4442:
4435:
4433:0-8176-3257-3
4429:
4425:
4421:
4417:
4412:
4408:
4406:0-534-37997-4
4402:
4398:
4393:
4389:
4387:0-471-30932-X
4383:
4379:
4375:
4371:
4369:
4367:
4363:
4359:
4358:
4353:
4349:
4345:
4341:
4335:
4331:
4326:
4322:
4320:0-12-059815-9
4316:
4312:
4307:
4306:
4293:
4291:981-02-4196-8
4287:
4283:
4282:
4274:
4266:
4264:9971-5-0290-9
4260:
4256:
4255:
4249:For example,
4246:
4238:
4236:0-8176-4199-8
4232:
4228:
4227:
4219:
4211:
4205:
4201:
4197:
4193:
4186:
4178:
4176:0-7923-5302-1
4172:
4168:
4164:
4160:
4154:
4146:
4140:
4136:
4132:
4125:
4117:
4111:
4107:
4103:
4097:
4093:
4087:
4085:3-540-26296-2
4081:
4077:
4073:
4066:
4058:
4056:0-8176-3715-X
4052:
4048:
4043:
4042:
4033:
4025:
4023:0-8176-3257-3
4019:
4015:
4011:
4006:
3997:
3989:
3983:
3979:
3978:
3971:
3967:
3966:Dirac algebra
3961:
3953:
3951:0-7923-2576-1
3947:
3943:
3939:
3935:
3928:
3921:
3915:
3907:
3905:0-486-63833-2
3901:
3897:
3896:
3888:
3882:
3877:
3875:
3873:
3871:
3856:(in Japanese)
3855:
3851:
3845:
3831:(in Japanese)
3830:
3826:
3820:
3812:
3810:981-02-4196-8
3806:
3802:
3801:
3793:
3789:
3780:
3777:
3774:
3773:Orientability
3771:
3768:
3765:
3763:
3760:
3758:
3755:
3754:
3748:
3746:
3741:
3739:
3736:-blade in an
3733:
3723:Note on usage
3720:
3714:
3709:
3705:
3686:
3680:
3677:
3674:
3669:
3653:
3652:
3651:
3649:
3648:
3639:
3632:
3625:
3621:
3601:
3587:
3566:
3555:
3540:
3539:
3538:
3536:
3535:
3515:
3507:
3496:
3490:
3486:
3480:
3476:
3472:
3467:
3463:
3457:
3453:
3448:
3444:
3439:
3428:
3422:
3418:
3412:
3408:
3404:
3399:
3395:
3389:
3385:
3380:
3376:
3371:
3360:
3354:
3350:
3344:
3340:
3336:
3331:
3327:
3321:
3317:
3312:
3308:
3300:
3288:
3287:
3286:
3281:
3274:
3267:
3260:
3252:
3245:
3236:
3230:
3221:
3213:
3209:
3205:
3204:
3199:
3194:
3190:
3185:
3166:
3161:
3150:
3144:
3140:
3134:
3130:
3126:
3121:
3117:
3111:
3107:
3102:
3098:
3093:
3082:
3076:
3072:
3066:
3062:
3058:
3053:
3049:
3043:
3039:
3034:
3030:
3025:
3014:
3008:
3004:
2998:
2994:
2990:
2985:
2981:
2975:
2971:
2966:
2962:
2954:
2942:
2941:
2940:
2933:
2928:
2927:basis vectors
2923:
2919:
2915:
2910:
2906:
2902:
2898:
2894:
2890:
2886:
2876:
2871:
2864:
2857:
2850:
2843:
2836:
2829:
2822:
2818:
2814:
2810:
2806:
2802:
2796:
2790:
2786:
2782:
2778:
2774:
2769:
2767:
2765:
2760:
2756:
2752:
2748:
2744:
2740:
2721:
2711:
2704:
2697:
2690:
2675:
2674:
2673:
2671:
2667:
2662:
2660:
2650:
2634:
2604:
2570:
2547:
2537:
2531:
2520:
2514:
2486:
2466:
2460:
2451:
2445:
2433:
2406:
2395:
2386:
2382:
2377:
2347:
2344:
2338:
2326:
2302:
2274:
2248:
2237:
2228:
2224:
2219:
2186:
2170:
2151:
2135:
2130:
2128:
2124:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2091:
2089:
2085:
2081:
2077:
2073:
2069:
2065:
2061:
2057:
2053:
2049:
2042:Formalization
2039:
2037:
2033:
2029:
2025:
2024:cross product
2021:
2017:
2013:
2003:
1994:
1988:
1985:
1982:
1979:
1978:
1977:
1972:
1953:
1935:
1926:
1917:
1896:
1878:
1869:
1860:
1842:
1836:
1818:
1812:
1808:
1792:
1788:
1772:
1768:
1748:
1747:
1746:
1741:
1737:
1729:
1722:
1715:
1707:
1700:
1693:
1688:
1683:
1676:
1638:
1620:
1611:
1602:
1584:
1578:
1560:
1550:
1549:
1548:
1546:
1537:
1528:
1526:
1522:
1515:
1494:
1489:
1473:
1469:
1452:
1448:
1444:
1440:
1424:
1415:
1395:
1375:
1346:
1345:
1344:
1339:
1332:
1313:
1292:
1283:
1265:
1262:
1244:
1235:
1226:
1208:
1202:
1198:
1182:
1178:
1162:
1158:
1138:
1137:
1136:
1131:
1127:
1119:
1112:
1105:
1097:
1090:
1083:
1078:
1072:
1049:
1031:
1022:
1013:
992:
974:
965:
956:
954:
931:
922:
913:
895:
886:
877:
875:
869:
853:
849:
833:
829:
805:
804:
803:
798:
794:
786:
779:
772:
764:
757:
750:
740:
738:
734:
695:
686:
677:
675:
669:
643:
640:
638:
632:
615:
614:
613:
610:
607:
604:
597:
591:
588:
581:
576:
571:
568:
561:
556:
552:
548:
544:
539:
537:
531:
529:
524:
523:contravariant
520:
516:
512:
507:
505:
501:
497:
493:
489:
485:
480:
477:
470:
465:
460:
457:
450:
445:
442:
438:
435:
431:
427:
423:
419:
413:
409:
399:
397:
393:
372:
368:
364:
359:
355:
351:
346:
342:
335:
320:
317:
313:
312:cross product
308:
306:
302:
296:
291:
286:
284:
278:
274:
270:
266:
256:
252:
250:
246:
242:
238:
234:
230:
220:
218:
214:
210:
209:
208:pseudotensors
204:
203:
202:pseudoscalars
198:
192:
187:
184:-dimensional
183:
179:
178:
173:
169:
164:
162:
161:cross product
158:
154:
150:
146:
141:
137:
132:
128:
124:
119:
117:
116:polar vectors
113:
109:
105:
101:
97:
93:
89:
85:
81:
77:
73:
69:
65:
57:
53:
49:
45:
42:
38:
35:
30:
24:
19:
4463:
4454:
4450:
4444:
4440:
4423:
4419:
4399:. Thompson.
4396:
4377:
4366:Axial vector
4365:
4356:
4329:
4313:. Harcourt.
4310:
4280:
4273:
4253:
4245:
4225:
4218:
4199:
4195:
4185:
4166:
4153:
4134:
4124:
4105:
4095:
4092:right-handed
4091:
4075:
4065:
4040:
4032:
4013:
4004:
3996:
3976:
3960:
3941:
3937:
3927:
3914:
3894:
3887:
3858:. Retrieved
3853:
3844:
3833:. Retrieved
3828:
3819:
3799:
3792:
3745:vector space
3742:
3737:
3731:
3726:
3712:
3707:
3703:
3701:
3645:
3637:
3630:
3623:
3619:
3616:
3532:
3530:
3279:
3272:
3265:
3258:
3250:
3243:
3234:
3228:
3219:
3211:
3207:
3201:
3192:
3188:
3181:
2931:
2921:
2917:
2913:
2908:
2904:
2900:
2896:
2893:pseudovector
2892:
2889:axial vector
2888:
2882:
2869:
2862:
2855:
2848:
2841:
2834:
2827:
2820:
2816:
2812:
2808:
2804:
2800:
2794:
2773:pseudovector
2772:
2770:
2763:
2758:
2754:
2750:
2736:
2669:
2663:
2656:
2131:
2126:
2122:
2110:
2094:
2092:
2087:
2083:
2079:
2075:
2067:
2063:
2060:pseudovector
2059:
2051:
2047:
2045:
2009:
2000:
1992:
1970:
1968:
1739:
1735:
1727:
1720:
1713:
1705:
1698:
1691:
1689:
1681:
1674:
1672:
1544:
1542:
1513:
1511:
1337:
1330:
1328:
1129:
1125:
1117:
1110:
1103:
1095:
1088:
1081:
1079:
1070:
1068:
796:
792:
784:
777:
770:
762:
755:
748:
746:
732:
730:
611:
605:
602:
595:
589:
586:
579:
574:
569:
566:
559:
554:
550:
540:
532:
514:
508:
503:
495:
491:
487:
483:
478:
475:
468:
463:
458:
455:
448:
443:
439:, so that a
436:
418:vector space
415:
321:
309:
300:
294:
287:
282:
276:
272:
268:
262:
226:
206:
200:
196:
190:
181:
175:
165:
157:vector field
139:
135:
130:
126:
120:
115:
76:axial vector
75:
72:pseudovector
71:
61:
55:
52:and reversed
51:
47:
43:
39:, creates a
36:
18:
4096:left-handed
2757:-values. A
2747:multivector
2739:dot product
2119:orientation
2115:volume form
2056:dimensional
1329:Therefore,
737:determinant
195:, written ⋀
155:of a polar
96:translation
68:mathematics
23:Free vector
4485:Categories
4302:References
3970:trivectors
3860:2023-11-07
3835:2023-11-07
3767:Antivector
3203:Hodge dual
2789:trivectors
2777:dimensions
2318:, one has
422:components
406:See also:
100:reflection
4380:. Wiley.
3678:−
3588:×
3556:∧
3473:−
3405:−
3337:−
3301:∧
3127:−
3059:−
2991:−
2955:×
2712:∧
2698:⋅
2571:ρ
2538:×
2521:≅
2434:ρ
2387:ρ
2327:ρ
2229:ρ
1897:×
1837:×
1793:×
1639:×
1579:×
1474:−
305:symmetric
245:vorticity
177:bivectors
84:direction
4376:(1999).
4161:(1999).
4010:: Duals"
3751:See also
3198:bivector
2785:bivector
2026:and the
1997:Examples
1809:′
1789:′
1769:′
1690:Suppose
1490:′
1470:′
1441:′
1199:′
1179:′
1159:′
870:′
850:′
830:′
747:Suppose
670:′
633:′
517:to be a
500:velocity
396:bivector
92:rotation
2562:. Then
2425:, with
2125:) with
1738:, then
1128:, then
795:, then
494:-, and
402:Details
64:physics
34:current
4470:
4430:
4403:
4384:
4336:
4317:
4288:
4261:
4233:
4206:
4173:
4141:
4112:
4098:; see
4082:
4053:
4020:
3984:
3948:
3902:
3807:
3684:
3617:where
3599:
3591:
3585:
3577:
3559:
3553:
3513:
2766:-blade
2719:
2575:pseudo
2438:pseudo
2391:pseudo
2117:or an
2050:is an
1673:where
519:tensor
515:vector
504:cannot
484:vector
283:actual
229:torque
217:tensor
213:scalar
80:vector
3785:Notes
2107:field
123:plane
4468:ISBN
4428:ISBN
4401:ISBN
4382:ISBN
4334:ISBN
4315:ISBN
4286:ISBN
4259:ISBN
4231:ISBN
4204:ISBN
4171:ISBN
4139:ISBN
4110:ISBN
4080:ISBN
4051:ISBN
4018:ISBN
3982:ISBN
3946:ISBN
3918:See
3900:ISBN
3805:ISBN
3734:– 1)
3706:and
3278:and
3208:dual
2907:and
2899:and
2891:and
2797:− 1)
2331:fund
2233:fund
2136:for
2099:even
2078:: ⋀(
2028:curl
1697:and
1680:and
754:and
410:and
316:curl
271:= Σ(
247:and
205:and
170:and
153:curl
129:and
74:(or
70:, a
66:and
4094:or
4047:234
3210:of
2875:}.
2873:123
2866:124
2859:134
2852:234
2657:In
2455:det
2287:in
2171:of
2109:of
2097:is
2074:of
2062:of
1969:So
1924:det
1867:det
1609:det
1527:.)
1290:det
1233:det
1069:So
1020:det
963:det
920:det
884:det
684:det
601:= −
530:.)
521:of
490:-,
297:= 0
215:or
193:− 1
118:).
90:by
62:In
48:not
4487::
4453:×
4443:∧
4426:.
4424:ff
4354:.
4350:.
4202:.
4200:ff
4194:.
4165:.
4133:.
4104:.
4074:.
4012:.
4005:Cℓ
3972:.
3944:.
3942:ff
3869:^
3852:.
3827:.
3636:∧
3629:∧
3622:=
3508:12
3440:31
3372:23
3249:∧
3223:23
3191:∧
2930:{
2920:×
2916:=
2868:,
2861:,
2854:,
2840:,
2833:,
2826:,
2768:.
2670:ab
2649:.
2598:SO
2525:SO
2129:.
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