Objective: Given two straight lines in 3D:

line 1: r = a + t*b

line 2: r = p +s*q
Show that the distance between those two lines are (a  p).u
where u is a unit vector given by b x q / b x q.
 Finding direction of the line segment
 Finding the length of the line segment that bridge line 1 and 2
To find the length, scalar projection can be applied. If we know a vector that bridges the two lines (in this case it is a  p), we can find the distance between those two lines by calculating scalar projection on the perpendicular vector. Remember that the distance is actually a  p cos theta where theta is an angle between (a  p) and the perpendicular vector.
The scalar projection is a  pcos theta =  (a  p) . (b x q)  /  b x q =  (a  p) . u.
Q.E.D.
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