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Since three or fewer points are always coplanar, the problem of determining when a set of points are coplanar is generally of interest only when there are at least four points involved. In the case that there are exactly four points, several
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1432:{\displaystyle {\begin{bmatrix}x_{1}-w_{1}&x_{2}-w_{2}&\dots &x_{n}-w_{n}\\y_{1}-w_{1}&y_{2}-w_{2}&\dots &y_{n}-w_{n}\\z_{1}-w_{1}&z_{2}-w_{2}&\dots &z_{n}-w_{n}\\\end{bmatrix}}}
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1148:{\displaystyle {\begin{aligned}X&=(x_{1},x_{2},\dots ,x_{n}),\\Y&=(y_{1},y_{2},\dots ,y_{n}),\\Z&=(z_{1},z_{2},\dots ,z_{n}),\\W&=(w_{1},w_{2},\dots ,w_{n}),\end{aligned}}}
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methods can be employed, but a general method that works for any number of points uses vector methods and the property that a plane is determined by two
525:{\displaystyle (\mathbf {c} \cdot \mathbf {\hat {a}} )\mathbf {\hat {a}} +(\mathbf {c} \cdot \mathbf {\hat {b}} )\mathbf {\hat {b}} =\mathbf {c} ,}
68:
provides a solution technique for the problem of determining whether a set of points is coplanar, knowing only the distances between them.
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are coplanar if and only if the matrix of their relative differences, that is, the matrix whose columns (or rows) are the vectors
35:, the plane they determine is unique. However, a set of four or more distinct points will, in general, not lie in a single plane.
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to this cross product through the initial point will lie in the plane. This leads to the following coplanarity test using a
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In the special case of a plane that contains the origin, the property can be simplified in the following way: A set of
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in three-dimensional space are coplanar if there is a plane that includes them both. This occurs if the lines are
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857:{\overrightarrow {p_{0}p_{1}}},\ {\overrightarrow {p_{0}p_{2}}},\ \dots ,\ {\overrightarrow {p_{0}p_{k-1}}}
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31:
that contains them all. For example, three points are always coplanar, and if the points are distinct and
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are not coplanar. Such a polygon must have at least four vertices; there are no skew triangles.
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points and the origin are coplanar if and only if the matrix of the coordinates of the
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vectors with the same initial point determine a plane through that point. Their
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1526:(Alternate ed.), Prindle, Weber & Schmidt, p.
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each other. Two lines that are not coplanar are called
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Geometric property of objects being in the same plane
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is of rank 2 or less, the four points are coplanar.
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247:{\displaystyle \cdot (x_{3}-x_{1})=0.}
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88:vector to that plane, and any vector
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871:For example, given four points
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72:Properties in three dimensions
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42:An example of coplanar points
635:linearly independent vectors
257:which is also equivalent to
27:if there exists a geometric
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1518:Swokowski, Earl W. (1983),
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644:-dimensional space where
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82:cross product
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54:, or if they
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33:non-collinear
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1494:Collinearity
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1463:skew polygon
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870:
868:2 or less.
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568:denotes the
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651:, a set of
570:unit vector
1559:"Coplanar"
1505:References
1478:polyhedron
90:orthogonal
60:skew lines
1564:MathWorld
1409:−
1394:…
1379:−
1354:−
1327:−
1312:…
1297:−
1272:−
1245:−
1230:…
1215:−
1190:−
1120:…
1055:…
990:…
925:…
851:→
842:−
812:…
801:→
766:→
715:−
698:…
574:direction
551:^
506:^
491:^
482:⋅
465:^
450:^
441:⋅
348:−
332:×
316:−
297:⋅
281:−
223:−
207:⋅
188:−
172:×
156:−
56:intersect
1577:Category
1488:See also
1471:vertices
52:parallel
25:coplanar
21:geometry
1467:polygon
1158:if the
655:points
572:in the
566:
537:
410:(i.e.,
1534:
1482:volume
1469:whose
1160:matrix
864:is of
818:
809:
774:
704:
695:
679:
640:In an
631:ad hoc
535:where
86:normal
1465:is a
84:is a
48:lines
29:plane
1532:ISBN
866:rank
598:and
416:and
46:Two
1528:647
649:≥ 3
604:on
592:on
586:of
576:of
408:= 0
19:In
1579::
1561:.
1530:,
1476:A
1461:A
637:.
616:.
404:⋅
394:,
390:,
370:0.
242:0.
121:,
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96::
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1088:=
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1001:n
997:y
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987:,
982:2
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958:=
951:Y
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941:)
936:n
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928:,
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896:(
893:=
886:X
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512:=
503:b
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488:b
478:c
474:(
471:+
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433:(
419:b
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324:1
320:x
311:4
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303:(
300:[
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272:x
268:(
239:=
236:)
231:1
227:x
218:3
214:x
210:(
204:]
201:)
196:1
192:x
183:4
179:x
175:(
169:)
164:1
160:x
151:2
147:x
143:(
140:[
126:4
123:x
119:3
116:x
112:2
109:x
105:1
102:x
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