Intersection parameters from scalar triple products ($s=[a,b,n]/[a,m,n],t=[a,b,m]/[a,m,n]$)

Consider lines $r=a+sm$ and $r=b+tn$ with nonzero direction vectors. If they intersect, then $a-b=tn-sm$, so the necessary coplanarity condition is $[a-b,m,n]=0$. For nonparallel directions it is also sufficient, since the orthogonal complement of $m\times n$ is the linear span of $m,n$. For parallel directions the condition alone is insufficient: distinct parallel lines also satisfy it.

Assume intersection and $D=[a,m,n]\ne 0$. Take the scalar triple products $[a,b,n]$ and $[a,b,m]$ of $b=a+sm-tn$. The scalar triple product is an alternating multilinear map, giving respectively $sD$ and $tD$, proving the title formulas. The nonzero denominator fixes a convenient determinant chart; intersections with $D=0$ can still exist and require a different auxiliary vector or another method.