If [( a b) ( b c) ( c a)]=[ a b c]^2, then is equal to:

Question

If $[(\vec a	imes\vec b)\ (\vec b	imes\vec c)\ (\vec c	imes\vec a)]=\lambda[\vec a\ \vec b\ \vec c]^2$, then $\lambda$ is equal to:

Accepted Answer

This is a standard identity: for any three vectors, $[(\vec a\times\vec b)\ (\vec b\times\vec c)\ (\vec c\times\vec a)] = [\vec a\ \vec b\ \vec c]^2$.

$$(\vec a\times\vec b)\cdot\big((\vec b\times\vec c)\times(\vec c\times\vec a)\big) = (\vec a\times\vec b)\cdot\big([\vec b\ \vec c\ \vec a]\vec c\big) = [\vec a\ \vec b\ \vec c]\big((\vec a\times\vec b)\cdot\vec c\big) = [\vec a\ \vec b\ \vec c]^2$$

So $\lambda=1$.

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