Consider the four points A = (1, 1, −1), B = (0, 1, 1), C = (1, −1, 0), and D =
( −2, 0, 3). Find the distance between the lines (AB) and (C D), that is, the distance between
the closest points on these two lines, using two different methods:
(a) Find a pair of planes that are parallel to both lines, with the first plane containing the line (AB) and the second containing the line (C D). Then find the distance between these two planes.
(b) Find parametric equations of the lines (AB) and (C D), and find the times at which the segment connecting a point P1 on (AB) to a point P2 on (C D) is perpendicular to both lines. The length of this segment is then the distance between the lines.
Important Note : since the two points P1 and P2 may be chosen independently of one another, the parameters of t1 and t2 of the two lines should be allowed to vary independently of one another. Thus there are two unknowns, and the condition that the the line segment P1P2 be perpendicular to the two lines gives you two equations, which you may solve.