Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 15 / d

Put $D=[a,m,n]\ne 0$ and rewrite the intersection equation as $b=a+sm-tn$. Take scalar triple products with $a$ and $n$:Taking the triple product with $a$ and $m$ instead givesThus the intersection parameters from scalar triple products areThe assumed nonzero denominator makes these expressions valid and guarantees nonparallel directions. The resulting point is $r=a+sm=b+tn$; these formulas are conditional on the intersection assumed in the question.