Shortest distance between $2$ lines with direction vectors

Question

Let L1 be the line passing through the point $P_1=(−11, 10, −1)$ with direction vector $\vec d_1=\begin{bmatrix}−2\\ 3\\ −1\end{bmatrix}t$, and let $L_2$ be the line passing through the point $P_2=(−8, 9, 10)$ with direction vector $\vec d_2=\begin{bmatrix}−1\\ 3\\1\end{bmatrix}t$. Find the shortest distance, $d$, between these two lines, and find a point $Q_1$ on $L_1$ and a point $Q_2$ on $L_2$ so that $d(Q_1,Q_2) = d$.

I dont really understand how to properly approach this question, how do i start?

Answer 1

Hints: any point on $L_1$ can be written as $(-11-2t,10+3t,-1-t)$ where t is a parameter. $Q_1$ is one such point. Likewise for $L_2$ with parameter, say s.

Note that the line connecting $Q_1$ and $Q_2$ is perpendicular to both $d_1$ and $d_2$ so, must be along their cross product $d_3$. Parametrise that direction by u. So,

$$Q_1 + u d_3=Q_2$$

The three components of this give 3 equations for s, t, u which are easily solved and you get everything you need.