Prove that if and are two planes which are not parallel then they intersect in a line.
Proof. Let the Cartesian equations of and be given by
respectively. Then, the intersection is given by the common solutions of these two equations. Since and are not parallel, we know they do not have the same normal vector so that for all . Further, since the normals are nonzero, we know each equation has at least one nonzero coefficient. Without loss of generality, let . Then,
Substituting into the Cartesian equation for we have
is the set of solutions for the points on . But, we know at least one of or is nonzero, otherwise . Hence, we have the equation for a line. Therefore, is a line