Consider two non-parallel straight lines in 3-dimensional space. The first 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]

/

square 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.