Problem source

Two non-parallel lines in 3-dimensional space are given by $\mathbf{r}=\mathbf{p}_{1}+t_{1}\mathbf{m}_{1}$
and $\mathbf{r}=\mathbf{p}_{2}+t_{2}\mathbf{m}_{2}$ respectively,
where $\mathbf{m}_{1}$ and $\mathbf{m}_{2}$ are unit vectors. Explain
by means of a sketch why the shortest distance between the two lines
is 
\[
\frac{\left|(\mathbf{p}_{1}-\mathbf{p}_{2})\cdot(\mathbf{m}_{1}\times\mathbf{m}_{2})\right|}{\left|(\mathbf{m}_{1}\times\mathbf{m}_{2})\right|}.
\]

\begin{questionparts}
\item Find the shortest distance between the lines in the case 
\[
\mathbf{p}_{1}=(2,1,-1)\qquad\mathbf{p}_{2}=(1,0,-2)\qquad\mathbf{m}_{1}=\tfrac{1}{5}(4,3,0)\qquad\mathbf{m}_{2}=\tfrac{1}{\sqrt{10}}(0,-3,1).
\]
\item Two aircraft, $A_{1}$ and $A_{2},$ are flying in the directions
given by the unit vectors $\mathbf{m}_{1}$ and $\mathbf{m}_{2}$
at constant speeds $v_{1}$ and $v_{2}.$ At time $t=0$ they pass
the points $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$, respectively.
If $d$ is the shortest distance between the two aircraft during the
flight, show that 
\[
d^{2}=\frac{\left|\mathbf{p}_{1}-\mathbf{p}_{2}\right|^{2}\left|v_{1}\mathbf{m}_{1}-v_{2}\mathbf{m}_{2}\right|^{2}-[(\mathbf{p}_{1}-\mathbf{p}_{2})\cdot(v_{1}\mathbf{m}_{1}-v_{2}\mathbf{m}_{2})]^{2}}{\left|v_{1}\mathbf{m}_{1}-v_{2}\mathbf{m}_{2}\right|^{2}}.
\]
\item Suppose that $v_{1}$ is fixed. The pilot of $A_{2}$ has chosen $v_{2}$
so that $A_{2}$ comes as close as possible to $A_{1}.$ How close
is that, if $\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{m}_{1}$ and $\mathbf{m}_{2}$
are as in \textbf{(i)}?
\end{questionparts}