Quick method to find line of shortest distance for skew lines

In linear algebra it is sometimes needed to find the equation of the line of shortest distance for two skew lines. What follows is a very quick method of finding that line.

Let’s consider an example. Start with two simple skew lines:

s : \begin{cases} x = 1 + t \\ y = 0 \\ z = -t \end{cases}

r : \begin{cases} x = - k \\ y = k + 2 \\ z = k \end{cases}

(Observation: don’t make the mistake of using the same parameter for both lines. Each lines exist on its own, there’s no link between them, so there’s no reason why they should should be described by the same parameter. If this doesn’t seem convincing, get two lines you know to be intersecting, use the same parameter for both and try to find the intersection point.)

The directional vectors are:

V_{s} = (1, 0, -1), V_{r} = (- 1, 1, 1)

So they clearly aren’t parallel. They aren’t incidental as well, because the only possible intersection point is for y = 0, but when y = 0, r is at (2, 0, -2), which doesn’t belong to s. It does indeed make sense to look for the line of shortest distance between the two, confident that we will find a non-zero result.

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