If a b= b c= c a, then a+ b+ c=

Question

If $\vec a	imes\vec b=\vec b	imes\vec c=\vec c	imes\vec a$, then $\vec a+\vec b+\vec c=$

Accepted Answer

Let $\vec r=\vec a+\vec b+\vec c$. Then $\vec a	imes\vec r = \vec a	imes\vec b+\vec a	imes\vec c = \vec a	imes\vec b-\vec c	imes\vec a=\vec 0$ (using $\vec a	imes\vec b=\vec c	imes\vec a$).

Similarly $\vec b	imes\vec r=\vec 0$ and $\vec c	imes\vec r=\vec 0$. These can hold simultaneously (for non-collinear $\vec a,\vec b,\vec c$) only if $\vec r=\vec 0$.

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