# Shortest distance between 2 lines in 3D

• Mar 9th 2010, 08:46 AM
DangerousDave
Shortest distance between 2 lines in 3D
Hi all,

I've been investigating the shortest distance between two lines in terms of vectors. So far I have that you need to find the common perpendicular, from which I can get two equations with three unknowns. I believe it's possible to find the ratio of the unknowns from this, but I'm not sure how to go about it. Any help appreciated.

Cheers
• Mar 9th 2010, 09:43 AM
Plato
Quote:

Originally Posted by DangerousDave
I've been investigating the shortest distance between two lines in terms of vectors. So far I have that you need to find the common perpendicular, from which I can get two equations with three unknowns. I believe it's possible to find the ratio of the unknowns from this, but I'm not sure how to go about it. Any help appreciated.

Suppose the we have two skew lines, $l_1 (t) = P + tD\quad \& \quad l_2 (t) = Q + tE$.
The distance between them is $\frac{{\left| {\overrightarrow {PQ} \cdot \left( {D \times E} \right)} \right|}}{{\left\| {D \times E} \right\|}}$
• Mar 9th 2010, 01:20 PM
DangerousDave
To put this into words, would this be the scalar product of (the line between a known point on each line) with (the unit vector for the scalar product of the direction vectors for each line)? I'm not sure of the significance of the 'x' signs.

Is there no merit in the other approach I mentioned?
• Mar 9th 2010, 01:30 PM
Plato
Quote:

Originally Posted by DangerousDave
I'm not sure of the significance of the 'x' signs.

Are you saying that you do not understand the idea of the cross product of two vectors?
If that is true what I posted is of absolutely no use to you.
• Mar 9th 2010, 01:35 PM
DangerousDave
I'm afraid so. I've finally got word back from the prof of my course and he says that we don't need to know it and haven't been taught it, even though it's on the official syllabus. I had wondered since, any way you cut it, it seems to involve mathematical techniques we haven't been taught yet.

Perhaps I'll investigate cross products anyway.

Thanks for the help.