Coplanarity

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In geometry, a set of points in space is coplanar if the points all lie in the same 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.

[edit] Properties

If three 3-dimensional vectors <math>\mathbf{a}, \mathbf{b} </math> and <math>\mathbf{c}</math> are coplanar, and <math>\mathbf{a}\cdot\mathbf{b} = 0</math>, then

<math>(\mathbf{c}\cdot\mathbf{\hat a})\cdot\mathbf{\hat a} + (\mathbf{c}\cdot\mathbf{\hat b})\cdot\mathbf{\hat b} = \mathbf{\hat c}, </math>

where <math>\mathbf{\hat a}</math> denotes the unit vector in the direction of <math>\mathbf{a}</math>.

Or, the vector resolutes of <math>\mathbf{c}</math> on <math>\mathbf{a}</math> and <math>\mathbf{c}</math> on <math>\mathbf{b}</math> add to give the original <math>\mathbf{c}</math>.