[Math] How come the cross product of two planes is collinear with the direction vector of the line

Question

If two planes intersect in a line, explain why the cross product of the normal vectors of the planes is collinear with the direction vector of the line.

Answer 1

Not only is a normal vector $\hat{n}$ perpendicular to every line in the plane $P$, but the converse is true as well: every line perpendicular to $\hat{n}$ is parallel to $P$.

So, if we take the cross product of $\hat{n}_1$ and $\hat{n}_2$, we get a vector perpendicular to both, which means it is parallel to both $P_1$ and $P_2$.