Coplanar vectors

Question

Prove that if $$\vec{a}\times\vec{b}+\vec{b}\times\vec{c}+\vec{c}\times\vec{a}=0$$ then $\vec{a},\vec{b},\vec{c} $ are coplanars.

One thing I know is that i have to get $\vec{a}\cdot(\vec{b}\times\vec{c})=0$ in any order, but I don't know how.

Answer 1

Hint: $\vec{a}\times \vec{b} + \vec{b}\times\vec{c} + \vec{c}\times\vec{a} = \vec{0}$ is the $\vec{0}$ vector, not a number. So if we take $\vec{a}\cdot LHS = \vec{a}\cdot RHS$, we get:

$\vec{a} \cdot \left(\vec{a}\times \vec{b} + \vec{b}\times\vec{c} + \vec{c}\times \vec{a} \right)= 0$

Now, what do you know about $\vec{a}\times\vec{b}$, $\vec{a}\times\vec{c}$ in terms of $\vec{a}$, and what would the LHS of the equation above become if we simplify it using what we know about the first and third term in parentheses? (LHS = left hand side, RHS = right hand side)