
Re: Distance Between Lines in R^3 (fwd)
Posted:
Jul 28, 2013 5:01 PM


In article <kste74$gue$1@news.albasani.net>, Thomas Nordhaus <thnord2002@yahoo.de> wrote:
> Am 26.07.2013 05:33, schrieb William Elliot: > > How do we find the shortest distance between two lines L, L' in R^3 ? > > > > 2. http://at.yorku.ca/cgibin/bbqa?forum=calculus;task=show_msg;msg=0792 > > > > How about doing it straight forward from scratch? Let P1(s) = v0 +s*v, > P2(t) = w0 + t*w be arbitrary points on the line L, L' resp. > ....
Yes. Then just make the vector (P1(s)  P2(t)) perpendicular to both lines:
v.((v0 +s*v)  (w0 + t*w)) = 0 w.((v0 +s*v)  (w0 + t*w)) = 0.
Therefore (v.v)s  (v.w)t =  v.v0 + v.w0 (w.v)s  (w.w)t =  w.v0 + w.w0.
Solve those linear equations for s and t, then find (P1(s)  P2(t)).
That's all. It's a traditional method in old textbooks which aren't read much any more, and it doesn't need any calculus.
Ken Pledger.

