Consider two non-parallel straight lines in 3-dimensional space. The rst line ca
ID: 2962832 • Letter: C
Question
Consider two non-parallel straight lines in 3-dimensional space. The 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
(b) Now suppose that the two lines are parallel. Find a new formula for D
Explanation / Answer
Let the lines be x = a + bu and y = c + dv. Points x and y have the shortest distance if the line connecting them is orthogonal to both lines. This results in two linear equations for u and v.
Line xy is orthogonal to line a + bu: (x-y)*b = 0, i.e. (a+bu-c-dv)*b = 0, or
(b*b)u - (b*d)v = (c-a)*b
Line xy is orthogonal to line c + dv: (x-y)*d = 0, i.e. (a+bu-c-dv)*d = 0, or
(b*d)u - (d*d)v = (c-a)*d
Solving the above two equations gives u,v and therefore the two points x and y.