Let $A, B, C, $and $D$ be four distinct points in $3-space$. If $AB×CD$ does not equal $0$ and $AC⋅(AB×CD)=0$, explain why the line through $A$ and $B$ must intersect the line through $C$ and $D$.

Could someone please explain the concept and ideas behind this question?


$$AB × CD \neq 0$$

That means $AB$ and $CD$ are not parallel.

$$AC ⋅ (AB ×CD)=0$$

That means $AC$ is perpendicular to the normal vector $ AB ×CD$, so $A,B,C,D$ must be coplanar. But we have shown that $AB$ is not parallel to $CD$, so the two line must intersect at some point on their plane


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