= Solution
At an intersection, the two <position> <vectors> agree, so $a-b=tn-sm$. The <scalar triple product> with $m,n$ is therefore
$$
[a-b,m,n]=t[n,m,n]-s[m,m,n]=\boxed{0},
$$
since a <determinant> with repeated columns vanishes. Geometrically the displacement of the two anchor points must lie in the same <plane> as the two <line> directions.
This is a necessary condition. If $m,n$ are not parallel, it is also sufficient, because their span is exactly the <plane> perpendicular to $m\times n$. For parallel directions the triple product vanishes automatically, even for distinct <lines>, so the condition alone cannot guarantee intersection.
Back to article page
