Equation of the Line of Intersection of Two Planes

Let $\vec r\cdot\vec n=p$ and $\vec r\cdot\vec m=q$ be two non-parallel planes, so $\vec n,\vec m$ are normal to them respectively. The line where they meet must be perpendicular to BOTH normals — hence it is parallel to $\vec n\times\vec m$.

To locate a specific point on that line, write the two planes in Cartesian form $a_1x+b_1y+c_1z=p$ and $a_2x+b_2y+c_2z=q$. The line of intersection meets at least one coordinate plane; for simplicity, suppose it meets $z=0$. Substituting $z=0$ into both plane equations gives a $2\times2$ linear system $a_1x+b_1y=p,\ a_2x+b_2y=q$; solving it gives one point $(x_1,y_1,0)$ on the line of intersection. …