Given 4 lines in Euclidean 3-space:

L_{1}: x = 1, y = 0;

L_{2}: y = 1, z = 0;

L_{3}: x = 0, z = 1;

L_{4}: x = y, y = -6z.

Find the equations of the two lines which both meet all of the L_{i}.

**Solution**

A routine computation. Assume the line meets L_{1} at (1,0,a) and L_{2} at (b,1,0). Then it is (x - 1) = t(x - b), y = t(y - 1), (z - a) = tz. So it can only cut L_{3} if 1/b = 1 - a, and L_{4} if 6a = 6ab - 1. This gives a quadratic for a, which we can solve to get a = -1/2 or 1/3. Hence the possible lines are (1,0,-1/2) + t(-1/3,1,1/2) and (1,0,1/3) + t(1/2,1,-1/3).

© John Scholes

jscholes@kalva.demon.co.uk

4 Sep 1999