coplanarity

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, left bracketing bar 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 right bracketing bar = 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 (x_3 - x_1)·[(x_2 - x_1)x(x_4 - x_3)] = 0.

parallel lines | plane | point-plane distance | skew lines