So we have two vectors in 3d space - they aren't parallel.

Line A goes through (1,2,7) and (2,3,4). Line B goes through (2,3,7) and (4,1,2).

I worked out the equations to be

r1=(1,2,7)+t(1,1,-3)

r2=(2,3,7)+s(2,-2,-5)

The question asks for me to find the shortest distance between these two lines... and it gives a hint: Find any vector that joins a point from one line to the other and then compute the scalar projection of this vector onto the vector orthogonal to both lines.

So I found a vector which I called PQ that goes through (1,2,7) and (2,3,7) which i get as (1,1,0).

For the vector that is orthogonal to both lines... I know I can use the cross product to find this but since I have parameters t and s, I'm not sure how to do it... Do I have to find the cross product between the direction of vectors r1 and r2?

Finally, I simply don't understand how the scalar projection of the vector PQ onto the vector perpendicular to both lines will give me the length. I understand the scalar projections in 2d vectors when they are tail to tail but the vectors PQ and the perpendicular vector aren't even touching, so how does that work?

Thanks in advanced!