= Solution
Put $D=[a,m,n]\ne0$ and rewrite the intersection equation as $b=a+sm-tn$. Take <scalar triple products> with $a$ and $n$:
$$
[a,b,n]=[a,a+sm-tn,n]=sD.
$$
Taking the triple product with $a$ and $m$ instead gives
$$
[a,b,m]=[a,a+sm-tn,m]=-t[a,n,m]=tD.
$$
Thus the <intersection parameters from scalar triple products> are
$$
\boxed{s=\frac{[a,b,n]}{[a,m,n]},\qquad t=\frac{[a,b,m]}{[a,m,n]}.}
$$
The 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.
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