Thread: Find the shortest distance

Find the shortest distance between the following pairs of nonparallel lines and find the points on the lines that are closest together.

The line connected the points where the lines are closest together must be perpendicular to both and so must lie in a plane that is perpendicular to both lines. Any plane perpendicular to the first line must be of the form x+ y- z= A for some constant A. Any plane perpendicular to the second line must be of the form x+ 2z= B for some constant B. Subtracting those two equations eliminates x and gives y- 3z= A- B, a constant, for all x and y in the plane. In particular, setting y= 1+ s and z= -1- s, as in the first line, gives 1+ s+ 3+ 3x= 4s+ 4= A- B. Setting y= 2, z= 2t as in the second line, gives 2- 6t= A- B. Putting those together, 4s+ 4= 2- 6t so that s= -(3t+ 1)/2.