Is the result of the actions $\left((\vec A+\vec B) \times (\vec A\times \vec B)\right)\cdot(\vec A \times \vec B)$ depends by $\vec A$ and $\vec B$

Question

I want to show that this action not depend by A and B vectors, I know that cross product of the same vector by itself is $0$. $$\left((\vec A+\vec B) \times (\vec A\times \vec B)\right)\cdot(\vec A \times \vec B)$$ I can use here in Associative property?
like : $$\left((\vec A \times \vec B) \times (\vec A\times \vec B)\right)\cdot (\vec A+ \vec B)$$ then its zero.
any suggestions? Thanks!

Answer 1

The vector cross product is not associative, but it does distribute over addition:

$$(A+B)\times (A \times B) = A \times (A \times B) + B \times (A \times B).$$

The dot product operation is commutative. So you might have some success in applying the property

$$\alpha\cdot (\beta \times \gamma) = \beta \cdot (\gamma \times \alpha) = \gamma \cdot (\alpha \times \beta).$$

Using this, let $C = A \times B$, and we find

$$\begin{align*} \left((A + B)\times (C)\right) \cdot (C) &= \left(A\times (C) + B \times (C)\right)\cdot (C) \\ &= (C)\cdot (A\times (C)) + (C)\cdot (B\times (C)) \\ &= A \cdot ((C)\times (C)) + B\cdot ((C)\times (C)) \\ &= 0. \end{align*}$$