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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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1443:{\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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1159:{\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
536:{\displaystyle (\mathbf {c} \cdot \mathbf {\hat {a}} )\mathbf {\hat {a}} +(\mathbf {c} \cdot \mathbf {\hat {b}} )\mathbf {\hat {b}} =\mathbf {c} ,}
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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
46:, 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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868:{\overrightarrow {p_{0}p_{1}}},\ {\overrightarrow {p_{0}p_{2}}},\ \dots ,\ {\overrightarrow {p_{0}p_{k-1}}}
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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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1537:(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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258:{\displaystyle \cdot (x_{3}-x_{1})=0.}
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570:{\displaystyle \mathbf {\hat {a}} }
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882:For example, given four points
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646:linearly independent vectors
268:which is also equivalent to
38:if there exists a geometric
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1529:Swokowski, Earl W. (1983),
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655:-dimensional space where
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44:non-collinear
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1505:Collinearity
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1474:skew polygon
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879:2 or less.
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579:denotes the
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662:, a set of
581:unit vector
1570:"Coplanar"
1516:References
1489:polyhedron
101:orthogonal
71:skew lines
1575:MathWorld
1420:−
1405:…
1390:−
1365:−
1338:−
1323:…
1308:−
1283:−
1256:−
1241:…
1226:−
1201:−
1131:…
1066:…
1001:…
936:…
862:→
853:−
823:…
812:→
777:→
726:−
709:…
585:direction
562:^
517:^
502:^
493:⋅
476:^
461:^
452:⋅
359:−
343:×
327:−
308:⋅
292:−
234:−
218:⋅
199:−
183:×
167:−
67:intersect
1588:Category
1499:See also
1482:vertices
63:parallel
36:coplanar
32:geometry
18:Coplanar
1478:polygon
1169:if the
666:points
583:in the
577:
548:
421:(i.e.,
1545:
1493:volume
1480:whose
1171:matrix
875:is of
829:
820:
785:
715:
706:
690:
651:In an
642:ad hoc
546:where
97:normal
1476:is a
95:is a
59:lines
40:plane
1543:ISBN
877:rank
609:and
427:and
57:Two
1539:647
660:≥ 3
615:on
603:on
597:of
587:of
419:= 0
30:In
1590::
1572:.
1541:,
1487:A
1472:A
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627:.
415:⋅
405:,
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253:0.
132:,
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107::
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1578:.
1462:k
1458:k
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1424:w
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969:=
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664:k
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633:n
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618:b
612:c
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514:b
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499:b
489:c
485:(
482:+
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458:a
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444:(
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424:a
417:b
413:a
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399:a
378:=
375:]
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367:1
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350:x
346:(
340:)
335:1
331:x
322:4
318:x
314:(
311:[
305:)
300:1
296:x
287:2
283:x
279:(
250:=
247:)
242:1
238:x
229:3
225:x
221:(
215:]
212:)
207:1
203:x
194:4
190:x
186:(
180:)
175:1
171:x
162:2
158:x
154:(
151:[
137:4
134:x
130:3
127:x
123:2
120:x
116:1
113:x
20:)
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