Intersection line of two planes

Two planes intersect by a straight line (we assume that and are not parallel). First let us write the equation of a straight line which passes through the point in the direction given by the vector . Since for each point on this straight line the vector we have

This is the vector equation for a straight line. Now, our line has to be perpendicular to and , thus we may choose . The point belongs two the both planes simultaneously, that is, and . Let us try to find is as . The conditions give

This set of equations always has a unique solution since .