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)*\left\{\begin{matrix}\mathbf u\\\mathbf v\\\mathbf w\end{matrix}\right\}=\mathbf0*\left\{\begin{matrix}\mathbf u\\\mathbf v\\\mathbf w\end{matrix}\right\}`

`\implies\begin{cases}\mathbf u*\mathbf u+\mathbf u*\mathbf v+\mathbf u*\mathbf w=\mathbf0\\\mathbf v*\mathbf u+\mathbf v*\mathbf v+\mathbf v*\mathbf w=\mathbf0\\\mathbf w*\mathbf u+\mathbf w*\mathbf v+\mathbf w*\mathbf w=\mathbf0\end{cases}`

`\implies\begin{cases}\mathbf u*\mathbf v+\mathbf u*\mathbf w=\mathbf0\\\mathbf v*\mathbf u+\mathbf v*\mathbf w=\mathbf0\\\mathbf w*\mathbf u+\mathbf w*\mathbf v=\mathbf0\end{cases}`

since
`\mathbf x*\mathbf x=\mathbf0` for any vector
`\mathbf x`.

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