Is this vector identity correct?

Question

I cannot remember where I have seen this formula. But it looks like this: $$ (\vec A \cdot \vec B) \vec C = (\vec C \otimes \vec A) \vec B $$ I think it is a strange formula since I cannot find it in any textbook I have read. So is it correct? How would one show it?

Answer 1

Use the bra-ket notation can help to see this. By this way you will have LHS $$ \big(\langle A| B\rangle\big)|C\rangle = |C\rangle \big(\langle A| B\rangle\big)\quad, $$ since $\langle A| B\rangle$ is a scalar. Then $$ |C\rangle \big(\langle A| B\rangle \big)= \big(|C\rangle \langle A|\big) |B\rangle\quad, $$ since vectors and co-vectors are associative. Then you will have $$ \big(|C\rangle \langle A|\big) |B\rangle = \big(\vec C \otimes \vec A\big) \vec B $$ as you want.

Answer 2

You may be thinking of $$ {\bf a}\cdot ({\bf b}\times {\bf c})= {\bf b}\cdot ({\bf c}\times {\bf a})={\bf c}\cdot ({\bf a}\times {\bf b}). $$