Let the lines have direction vector
which I'll denote by r
, and without loss of generality let |r|
= 1 so a^2 + b^2 + c^2 = 1;
b = 1 - a; c = 1 - a so
The solutions are a =1, b = 0, c = 0 and a = 1/3, b = 2/3, c = 2/3.
(i) A has position vector
Let P =
, and Q =
AQ . BP
= 0 so
. Showing that the latter is > 0; 1 - sqrt(6)/3 > 0 therefore 1 > sqrt(6)/3 therefore 1 > 6/9, which is true.
The second equation gives
. Substituting into the third gives
. Substituting these into the first:
; multiplying out gives
, so no non-zero solutions.