Shortest distance between two lines and find points on each line I'm given the following two lines:
$L_1$: $P_1=(−13, 3, 14)$ with direction vector $d_1=(2, −1, −2)$
$L_2$: $P_2=(5, 4, 4)$ with direction vector $d_2=(−2, 1, 0)$
I'm then asked to find the shortest distance $d$ between these two lines, and then find a point, $Q_1$, on $L_1$, and a point, $Q_2$, on $L_2$ so that $d(Q_1,Q_2) = d$.
So far, I've determined that the shortest distance between these two points can be solved with a projection of the vector $\vec{P_1P_2}$ onto the direction vector found by the cross product of $d_1$ and $d_2$. In this case, it is $(4,8,0)$, or a magnitude of $4\sqrt{5}$. 
I'm not really sure how to determine the two points now that I've gotten the distance, any tips or explanations would be appreciated.

Solve the subproblem ("what is the distance from a point to a line?" (you should have this in your notes or book)) then perform a constrained optimization problem by allowing that point to vary over the line.
