If $u$ and $v$ are vectors in $R^3$, simplify the expression $(u+v) \times (u-v)$ as much as possible.

Question

Here is my thinking process for answering this question:

Cross product is neither commutative nor associative. Hence I cannot do any algebraic operations on this expression. However I know that cross product of two vectors is perpendicular to both of them.

On the other hand I know that $u-v = u + (-v)$ Therefore $u+v$ and $u-v$ are located are collinear with each other.

How can I use this information to simplify this expression?

Answer 1

$$\begin{align*} &(u +v) \times (u - v)\\ =& u \times u + v \times u - u \times v - v \times v\\ =& 0 + v \times u - u \times v - 0\\ =& v \times u - u \times v\\ =& v \times u + v \times u\\ =& 2 (v \times u) \end{align*}$$