Distance between two skew lines - why do we need the projection?

Question

https://i.stack.imgur.com/zZceh.jpg

I've been trying to understand that after getting vector n (the one derived from the cross product of the two direction vectors). If that vector connects both of them, why don't we just use the magnitude of n? Why do we need to do the scalar projection of PQ onto n?

I have a midterm today and I'd really appreciate a simple explanation, I'm sure it's just something I'm overthinking

Thanks!

Answer 1

The vector $\vec n$ could be any length $-$ it depends on the particular parametrisations of the lines. Only its direction is independent of the parametrisation. So you divide it by its magnitude to get a unit vector, $\vec u$ say. Then you want the projection of $\vec a$ onto $\vec u$, where $\vec a$ can be any line segment from $\vec r$ to $\vec p$.