### 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

.