= Intersection parameters from scalar triple products
{title2=$s=[a,b,n]/[a,m,n],\quad t=[a,b,m]/[a,m,n]$}
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 $D=[a,m,n]\ne0$. 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.
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