# Distance between two lines in R^3

1. Oct 19, 2013

### SteveDC

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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?

2. Oct 19, 2013

### Simon Bridge

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

3. Oct 19, 2013

### 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?

4. Oct 19, 2013

### D H

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?

5. Oct 19, 2013

### 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.

6. Oct 19, 2013

### D H

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.

7. Oct 19, 2013

### HallsofIvy

Staff Emeritus
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.

8. Oct 19, 2013

### D H

Staff Emeritus
That's an alternative approach, Halls, that isn't quite pertinent to the problem at hand. The OP was asked to use the normal to both lines and a line segment connecting a point on one line to a point on the other line.