Question

If u+v+w=0, show that u x v = v x w = w x u.

Answer 1

Take the cross product of both sides of $\mathbf u+\mathbf v+\mathbf w=\mathbf0$ with $\mathbf u,\mathbf v,\mathbf w$ to generate a system of three simultaneous equations.

$(\mathbf u+\mathbf v+\mathbf w)\times\left\{\begin{matrix}\mathbf u\\\mathbf v\\\mathbf w\end{matrix}\right\}=\mathbf0\times\left\{\begin{matrix}\mathbf u\\\mathbf v\\\mathbf w\end{matrix}\right\}$
$\implies\begin{cases}\mathbf u\times\mathbf u+\mathbf u\times\mathbf v+\mathbf u\times\mathbf w=\mathbf0\\\mathbf v\times\mathbf u+\mathbf v\times\mathbf v+\mathbf v\times\mathbf w=\mathbf0\\\mathbf w\times\mathbf u+\mathbf w\times\mathbf v+\mathbf w\times\mathbf w=\mathbf0\end{cases}$
$\implies\begin{cases}\mathbf u\times\mathbf v+\mathbf u\times\mathbf w=\mathbf0\\\mathbf v\times\mathbf u+\mathbf v\times\mathbf w=\mathbf0\\\mathbf w\times\mathbf u+\mathbf w\times\mathbf v=\mathbf0\end{cases}$

since $\mathbf x\times\mathbf x=\mathbf0$ for any vector $\mathbf x$.

Finally, use the fact that the cross product is anticommutative, i.e. $\mathbf x\times\mathbf y=-\mathbf y\times\mathbf x$. The conclusion follows.