Take the cross product of both sides of [tex]\mathbf u+\mathbf v+\mathbf w=\mathbf0[/tex] with [tex]\mathbf u,\mathbf v,\mathbf w[/tex] to generate a system of three simultaneous equations.
[tex](\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\}[/tex]
[tex]\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}[/tex]
[tex]\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}[/tex]
since [tex]\mathbf x\times\mathbf x=\mathbf0[/tex] for any vector [tex]\mathbf x[/tex].
Finally, use the fact that the cross product is anticommutative, i.e. [tex]\mathbf x\times\mathbf y=-\mathbf y\times\mathbf x[/tex]. The conclusion follows.
