Given a point A and a line
r=r0+tm,
we can find the shortest distance from the point to the line easily with
d=∣m∣∣∣AB×m∣∣.
Given two parallel planes with a normal vector
n
(it doesn’t matter which plane you take it from) and containing points A and B, we can find the shortest distance between them using the scalar projection:
d=∣∣ABn∣∣=∣∣AB⋅n^∣∣=∣n∣∣∣AB⋅n∣∣.
(Note that nonparallel planes intersect, so the shortest distance is zero.) We can also use this equation for skew lines, where one line contains A and the other has B. We calculate
n
by crossing their direction vectors: