Simplify expression cross/dot products

Question

Can the following expression be simplified? In particular, I would like to remove the denominator:

$\frac{( b \times (a \times b)) \cdot (b \times (a\times b))}{(a\times b) \cdot (a\times b)}$

where $a$ and $b$ are 3 dimensional vectors.

Answer 1

The expression is $\|b\|^2$, because $b$ and $c:=a\times b$ are orthogonal and $\|b\times c\|=\|b\|\|c\|.$