How to simplify $(\textbf{a} - \textbf{b})\times(\textbf{a}+\textbf{b})$?

Question

Are there any interesting ways of simplifying $(\textbf{a} - \textbf{b})\times(\textbf{a}+\textbf{b})$, where $\textbf{a}$ and $\textbf{b}$ are 3D vectors before I begin to dismantle the expression into a mess like a surgeon?

Thanks in advance.

HMA

Answer 1

$(\mathbf a-\mathbf b)\times(\mathbf a+\mathbf b)$

$=(\mathbf a-\mathbf b)\times \mathbf a+(\mathbf a-\mathbf b)\times \mathbf b $

$= \mathbf a\times \mathbf a-\mathbf b\times \mathbf a+\mathbf a\times \mathbf b-\mathbf b\times \mathbf b$

$=\mathbf 0+\mathbf a\times \mathbf b + \mathbf a\times \mathbf b - \mathbf 0$

$=2 \;\mathbf a\times \mathbf b$

by distributive, anti-commutative, and self cross product properties.