Coplanar -- From Wolfram MathWorld

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Coplanar

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Geometric objects lying in a common plane are said to be coplanar. Three noncollinear points determine a plane and so are trivially coplanar. Four points are coplanar iff the volume of the tetrahedron defined by them is 0,

|x_1 y_1 z_1 1; x_2 y_2 z_2 1; x_3 y_3 z_3 1; x_4 y_4 z_4 1|=0.

Coplanarity is equivalent to the statement that the pair of lines determined by the four points are not skew, and can be equivalently stated in vector form as

An arbitrary number of points , ..., can be tested for coplanarity by finding the point-plane distances of the points , ..., from the plane determined by and checking if they are all zero. If so, the points are all coplanar.

A set of vectors is coplanar if the nullity of the linear mapping defined by has dimension 1, the matrix rank of (or equivalently, the number of its singular values) is (Abbott 2004).