Question

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

Answer 1

$\mathbf u+\mathbf v+\mathbf w=\mathbf 0$

Take the cross product of both sides with $\mathbf u,\mathbf v,\mathbf w$, respectively. By the distributive property of the cross product, you have the system

$\begin{cases}\mathbf u\times\mathbf u+\mathbf u\times\mathbf v+\mathbf u\times\mathbf w=\mathbf 0\\\mathbf v\times\mathbf u+\mathbf v\times\mathbf v+\mathbf v\times\mathbf w=\mathbf 0\\\mathbf w\times\mathbf u+\mathbf w\times\mathbf v+\mathbf w\times\mathbf w=\mathbf 0\end{cases}\iff\begin{cases}\mathbf u\times\mathbf v+\mathbf u\times\mathbf w=\mathbf 0\\\mathbf v\times\mathbf u+\mathbf v\times\mathbf w=\mathbf0\\\mathbf w\times\mathbf u+\mathbf w\times\mathbf v=\mathbf 0\end{cases}$

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

And because the cross product is anticommutative, you have

$\begin{cases}\mathbf u\times\mathbf v-\mathbf w\times\mathbf u=\mathbf 0\\-\mathbf u\times\mathbf v+\mathbf v\times\mathbf w=\mathbf0\\\mathbf w\times\mathbf u-\mathbf v\times\mathbf w=\mathbf 0\end{cases}$

from which it follows immediately that

$\mathbf u\time\mathbf v=\mathbf v\times\mathbf w=\mathbf w\times\mathbf u$