Perhaps closely related is the problem of the shortest distance between two non-parallel ("skew") lines in 3 (or more) dimensions. I googled this & found this wikipedia entry: http://en.wikipedia.org/wiki/Skew_lines I'm not happy about the ugliness of these expressions. Yes, the "triple product" is a bit more satisfying, but I seem to recall seeing an expression somewhere that was much more satisfying -- something along the lines of the problem of the "distance of a point to a line", where the distance is obtained by plugging the point into the standard form of the line in terms of direction cosines: directed distance from point (x0,y0) to sin(theta)*x+cos(theta)*y+C is sin(theta)*x0+cos(theta)*y0+C. http://www.intmath.com/plane-analytic-geometry/perpendicular-distance-point-... I checked about 10 pages of Google's results, but couldn't find anything better than the wikipedia-type calculation for skew lines. At 11:29 AM 3/19/2011, Fred lunnon wrote:
Given non-parallel lines L,M in Euclidean 3-space, there exist unique planes S,U meeting in L, and T,V in M, with both S,T perpendicular to both U,V.
An algebraic proof of this turned out to be surprisingly nontrivial --- is there a more intuitive synthetic demonstration?
[The appropriate generalisation of this this apparently tedious little lemma provides one crucial link in the classification of isometries for a quadratic inner-product space. It may well be a (special case of) some well-known theorem concerning bases of vector subspaces, though I didn't recognise it as such.]
Fred Lunnon