Suppose we have two intersecting planes, P1 and P2, whose normal vectors are n1 and n2, respectively. If one was asked to find the line of intersection, one could simply calculate $\displaystyle \vec{n}_1 \times \vec{n}_2 = \vec{u}$, of which $\displaystyle \vec{u}$ would be the direction vector for the line.

My question is, why is it the case that calculating the cross product of the normal vectors of the planes provides the direction vector?