Question

If 3 vectors $\overline{a}.\overline{b},\overline{c}$ all lie in one plane (i.e., they are coplanar) then $\overline{c}$. $(\overline{a}\times \overline{b})$ = _______
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Answer 1

The definition of cross product says that not only $(\overline{a}\times \overline{b})$ is a vector perpendicular to both $\overline{a}$ and $\overline{b}$ but it is also perpendicular to the plane containing $\overline{a}$ and $\overline{b}$. So $\overline{a}$ $\times$ $\overline{b}$ is perpendicular to every vector in the plane containing $\overline{a}$ and $\overline{b}$. Since $\overline{c}$ lies in the same plane as of $\overline{a}$ and $\overline{b}$, that means c is perpendicular to $\overline{a}$ and $\overline{b}$, and we know that the dot product of two perpendicular vectors is always 0.
So $\overline{c}$. $(\overline{a}\times \overline{b})$ = 0.