If a b=1, b c=2, c a=3, then a( b c)+ b( c a)+ c( a b) is equal to:

Question

If $\vec a\cdot\vec b=1$, $\vec b\cdot\vec c=2$, $\vec c\cdot\vec a=3$, then $\vec a	imes(\vec b	imes\vec c)+\vec b	imes(\vec c	imes\vec a)+\vec c	imes(\vec a	imes\vec b)$ is equal to:

Accepted Answer

By the vector (Jacobi-type) identity, for any three vectors:
$$\vec a\times(\vec b\times\vec c)+\vec b\times(\vec c\times\vec a)+\vec c\times(\vec a\times\vec b) = \vec 0$$

This holds identically (verified directly: each term expands via $(\vec P\cdot\vec R)\vec Q-(\vec P\cdot\vec Q)\vec R$-type identities, and the three sums of coefficients cancel), regardless of the specific dot-product values given. So the answer is $\vec 0$.

Solutions · 1