Date: Jul 26, 2013 1:54 AM
Author: quasi
Subject: Re: Distance Between Lines in R^3  (fwd)

William Elliot wrote:

>How do we find the shortest distance between two lines L,L' in R^3?
>
>http://at.yorku.ca/cgi-bin/bbqa?forum=calculus;task=show_msg;msg=0792


Showing a reference is good, but at least give the right link.

The correct link is:

<http://at.yorku.ca/cgi-bin/bbqa?forum=calculus;task=show_msg;msg=0793>

A question of that type can typically be found as a worked example
in a Calculus 3 (Multivariate Calculus) textbook.

One way to do it is as follows ...

Denote the lines as L1,L2.

(1) Let v1,v2 be direction vectors for L1,L2 respectively.

(2) Compute the vector n = v1 x v2 (the cross product).

(3) Take arbitrary points P1,P2 on L1,L2 respectively.

(4) Get an equation for the plane through the point P1 with
normal vector n.

(5) Regarding P1,P2 as vectors, compute the vector difference
w = P2 - P1.

(6) Let p be the vector projection of w onto n.

(7) The distance between the lines L1,L2 is just the norm of p.

quasi