Select four points

,

,

,

in three dimensions to determine two lines

(green) and

(blue). For almost all choices of coordinates, the lines are

*skew*: neither parallel nor intersecting. The goal is to find the shortest distance

between the two lines. As a by-product, the shortest segment between the lines is shown in red.

Write the points as vectors

,

,

,

.

The cross-product of the directions of the lines is

, which gives a vector perpendicular to both lines. The unit vector in this direction is then

. The equations

and

determine parallel planes through the two lines at distances

and

from the origin. Therefore,

.

Here is one way to find the shortest line segment

connecting the two lines. Project the points

and

to the plane containing the line

; call the projected points

and

. Let

be the intersection of

and

.

To find the intersection, express the lines in parametric form:

and

. Set the right-hand sides equal and solve for

. Substitute that value of

in either equation to find

.

Similarly for

: Project the points

and

to the plane containing

; call the projected points

' and

. Let

be the intersection of

and

.