How come that when we want to prove that two lines are skew (that is that they don't intersect nor that they are parallel) we show that $C:=\vec{V_{1}} \times \vec{V_{2}} \cdot \overrightarrow{M_{1}M_{2}}\neq0 $? I understand that the main intent is to show that the parallelepiped these vectors create has volume, but still it is somewhat vague to me. ($V_{1} , V_{2}$ direction vectors, $M_{1},M_{2}$ dots on the two lines).
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To prevent this question from going to waste in the unanswered queue:
- If the two lines are parallel, then $\vec {V_1}\times \vec{V_2}=0\implies C= 0$.
- If the two lines intersect, then $\vec{V_1}\times \vec{V_2}$ is perpendicular to $\overrightarrow{M_1M_2}$ $\implies C= 0$.
- If the two lines are skew, then $\vec{V_1}\times\vec{V_2}$ is nonzero and not perpendicular to $\overrightarrow{M_1M_2}$ $\implies C\neq 0$.