Question: Two lines, l1 and l2, have equations r = -2i + k + t(i + j + k) and r = i + 5j + 2k + s(i + 2j + 3k) where s and t are constants. Points P and Q lies on the lines l1 and l2 respectively. If PQ is perpendicular to both l1 and l2, find the coordinates of P and Q and hence show that the length of PQ is √6 .

This is how I solve the problem:

l1 : r = (t - 2) i + t j + (t + 1)k

l2 : r = (1 + s )i + (5 +2s) j + (2 + 3s) k

Let b1 = i + j + k

Let b2 = i + 2j + 3k

b1 × b2 = i – 2j + k

Since line b1 × b2 is perpendicular to l1 and l2 , then I try to use the scalar product of (b1 × b2 ) ∙ P and (b1 × b2 ) ∙ Q to get the constants s and t , but I failed, could anyone help me to solve this problem? Thanks.