# Distance between two lines in R^3

SteveDC

## Homework Statement

Let line L1 be the line through origin O and the point B(2,-1,-2). Let line L2 be the line through point A(3,1,-1) with direction vector i-j-k. Obtain a vector n, orthogonal to both lines. Use n and the vector OA connecting the lines to find the shortest distance between two lines.

## The Attempt at a Solution

I have used both direction vectors for both lines to find their cross product to obtain the vector n, orthogonal to both lines: (2i-j-2k) x (i-j-k) = -i-k

But then not sure where to go from here despite the hint in the question of using vector OA and n?

Homework Helper
If you drew a line-segment joining the closest points on the lines to each other, what sort of properties would it have?

SteveDC
It would have direction n and have two points, one lying on L1 and one lying on L2, but I can't think of a way of finding these points?

Staff Emeritus
Given some point p1 on line 1 and some point p2 on line 2, what is the projection of the vector from point p1 to point p2 onto the vector n that is normal to both lines?

SteveDC
Am I right in saying that vector p1>p2 projected onto n would always be the distance between the two lines, but even if this is the case, how would you find the smallest value for this to find the shortest distance.

Staff Emeritus
Write an expression for some arbitrary point p1 on line 1 and some arbitrary point p2 on line 2. What is the projection of p2-p1 onto a vector perpendicular to both lines?

It's perhaps better if you any two arbitrary lines rather than your specific lines because the result is very general.

Hint: Use ##\vec p_i + \lambda\vec \tau_i## as the equation of a line passing through some point pi.

The line of "shortest distance" between two lines is, as said above, perpendicular to both lines. Find the plane perpendicular to the first line at an arbitrary point $(x_0, y_0, z_0)$. Determine the point, $(x_1, y_1, z_1)$, where the second line crosses that plane. Determine what $(x_0, y_0, z_0)$ must be so the line between $(x_0, y_0, z_0)$ and $(x_1, y_1, z_1)$ is also pependicular the second line.