Shortest Distance between 2 straight lines

May 2011
Consider two non-parallel straight lines in 3-dimensional space. The fi rst line can be
described, in Cartesian coordinates (x; y; z), by the parametric equations
x(u) = x1 + ua1 ; y(u) = y1 + ub1 ; z(u) = z1 + uc1
for some set of numbers (x1; y1; z1) and (a1; b1; c1). Likewise the second line can described
by x(v) = x2 + va2 ; y(v) = y2 + vb2 ; z(v) = z2 + vc2
for another set of numbers (x2; y2; z2) and (a2; b2; c2).

(a) Verify that the shortest distance D between this pair of lines is given by
D =

[x1 x2 y1 y2 z1 z2]
[a1 b1 c1]
[a2 b2 c2]
s quare root
([a1 b1]
[a2 b2])^2


([b1 c1]
[b2 c2])^2


([c1 a1]
[c2 a2])^2

Be sure to point out where in your analysis you use the assumption that the lines are not parallel.
(b) Now suppose that the two lines are parallel. Find a new formula for D.
Apr 2005
Don't you think it would be a good idea to show us what you have done on this problem?

Also, your notation is almost impossible to understand. Are those arrays supposed to be matrices?
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