Three vectors a, b, c are coplanar iff [a b c] = 0. Equivalently, one of them can be written as a linear combination of the other two: c = pa + qb.

Four points A, B, C, D are coplanar iff [AB AC AD] = 0. This is equivalent to saying the three vectors from one vertex to the other three are coplanar.

Two lines r=a1+tb1 and r=a2+sb2 are coplanar iff [a2-a1, b1, b2] = 0. If b1 x b2 != 0 (non-parallel), coplanar lines must intersect. If b1 x b2 = 0 (parallel), they are automatically coplanar.

To check if n vectors are linearly dependent: for 3D vectors, any set of 4+ vectors is always linearly dependent. Three vectors are linearly dependent iff their STP is zero.