quasi
Posts:
10,967
Registered:
7/15/05


Re: Distance Between Lines in R^3 (fwd)
Posted:
Jul 26, 2013 1:54 AM


William Elliot wrote:
>How do we find the shortest distance between two lines L,L' in R^3? > >http://at.yorku.ca/cgibin/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/cgibin/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

