Using index notation to show these equations are equal

Question

$$({\bf a}\times {\bf b})\cdot ({\bf a}\times {\bf b}) = |{\bf a}|^2|{\bf b}|^2 - ({\bf a}\cdot {\bf b})$$

I am doing the question above am stuck pretty much at the beginning. Any help? Thanks!

Answer 1

HINT:

Call ${\bf a} = a_i{\bf e}_i$, and ${\bf b}=b_j{\bf e}_j$ and remember that

$$ {\bf a}\times {\bf b} = {\bf e}_i\epsilon_{ijk}a_jb_k $$

You may also want to use the expression

$$ \epsilon_{ijk}\epsilon_{mnk} = \delta_{im}\delta_{jn}-\delta_{in}\delta_{jm} $$