Coplanarity
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In geometry, a set of points in space is coplanar if the points all lie in a geometric plane. For example, three points are always coplanar; but four points in space are usually not coplanar.
Points can be shown to be coplanar by determining that the scalar product of a vector that is normal to the plane and a vector from any point on the plane to the point being tested is 0.
Distance geometry provides a solution to the problem of determining if a set of points is coplanar, knowing only the distances between them.
Properties
If three 3-dimensional vectors [\mathbf, \mathbf ] and [\mathbf] are coplanar, and [\mathbf\cdot\mathbf = 0], then
- [(\mathbf\cdot\mathbf)\cdot\mathbf + (\mathbf\cdot\mathbf)\cdot\mathbf = \mathbf, ]
Or, the vector resolutes of [\mathbf] on [\mathbf] and [\mathbf] on [\mathbf] add to give the original [\mathbf].
External link
- , [Coplanar] at MathWorld.
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