Coplanarity

1) Three vectors are coplanar if one can be expressed as a linear combination of the other two.
2) Three vectors  , are coplanar if the points A, B, C and D are in the same plane

3) If three vectors a,b,c are coplanar, then a.(b x c) = 0

This scalar triple product can also be understood as the determinant of the 3x3 matrix having the three vectors as its rows and columns (the determinant of a transposed matrix is the same as the original); this quantity is invariant under coordinate rotation.