Let p1 and p2 be the two parallel planes. Let n1 be the normal vector of plane p1 (which is a vector perpendicular to the plane). If p2 is parallel to p1, then n1 is also a normal vector for p2.
Let p3 be the third secant plane, and n3 be its normal vector.
The direction vector of the intersection line of two planes is given by the cross products of their normal vectors (this is due to the fact that the cross product of two vectors is orthogonal two both of them, and that the direction vector of the intersection must be orthogonal two both normal vectors). So, the direction vectors of the two lines are:
