Show that if $\textbf{u}+\textbf{v}+\textbf{w}=\textbf{0}$, then $\textbf{u}\times\textbf{v}=\textbf{v}\times\textbf{w}=\textbf{w}\times\textbf{u}$

Question

This question doesn't really make sense to me unless the vectors are all the zero vector. So many things about this don't make sense to me that I fear writing them all down would confuse the reader, so I won't. I'll simply ask: how should I think about this problem? A solution would also be nice.

Answer 1

If $\bf u+v+w = 0$ then $\bf u,v,w$ all lie in the same plane, and if placed head to tail will form a triangle.

$\bf u\times v$ has a direction perpendicular to the plane holding $\bf u,v$ and magnitude equal to the area of the parallelogram formed by $\bf 0, u,v, u+v.$ Which will be twice the area of the triangle formed placing $\bf u,v,w$ into the triangle described above.

Any combination of $\bf u\times v, v\times w, w\times u $ will have similar direction and magnitude. However $\bf v\times u$ will have equal magnitude but opposite direction.

Answer 2

If $u+v+w=0$ then $w=-u-v$ and so $v\wedge w=v\wedge(-u-v) =-v\wedge u-v\wedge v$ etc.

Answer 3

This question doesn't really make sense to me unless the vectors are all the zero vector.

Let me provide you with another two examples of vectors $u,v,w$ such that $u+v+w=0$. Consider $u = (1,0,0)$, $v = (-1,0,0)$, and $w=0$. Also consider $u=(1,0,0)$, $v=(0,1,0)$, and $w=(-1,-1,0)$. Now the question should make more sense.

how should I think about this problem?

Think about examples like the ones I gave, and try to see whether the conclusion holds, and why. Once you're comfortable with examples you can move on to thinking about the general case and how to manipulate the equations.

Answer 4

Because $$0=u\times(u+v+w)=u\times v+u\times w=u\times v-w\times u.$$ The rest is similar.

Answer 5

Hint

$$\textbf{u}\times\textbf{v}-\textbf{v}\times\textbf{w}=\textbf{u}\times\textbf{v}+\textbf{w}\times\textbf{v}=-\textbf{v}\times\textbf{v}\\ \textbf{v}\times\textbf{w}-\textbf{w}\times\textbf{u}=\textbf{v}\times\textbf{w}+\textbf{u}\times\textbf{w}=-\textbf{w}\times\textbf{w}\\$$