I don't know what the author of the questions wants you to do.
But lines $L_1~\&~L_2$ are skew. It is well known that the is a unique perpendicular to each of two skew lines.
Suppose that $L_1: A+tD~\&~L_2: B+sE$
Using the hint, let $P\in L_1~\&~Q\in L_2$ So $\overrightarrow {PQ}$ will perpendicular to both $D~\&~E$.
Moreover, $\overrightarrow {PQ}$ will parallel to $D\times E$.
This is quite a tedious task. I don't envy you trying to juggle these ordinates.