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,

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).