So in certain papers I have seen people use the following vector identity:
$\hat{a} \times (\hat{a} \times \hat{c} ) = - \hat{c}$
where $\hat{a}$ and $\hat{b}$ are unit vectors. Intuitively this makes sense. However I need to prove it to make sure I'm correct. I know the vector triple product is given as:
$\hat{a} \times (\hat{b} \times \hat{c} ) = \hat{b} ( \hat{a} \bullet \hat{c} ) - \hat{c} ( \hat{a} \bullet \hat{b} ) $
now assuming $\hat{a} = \hat{b}$ we get something to the effect of
$\hat{a} \times (\hat{a} \times \hat{c} ) = \hat{a} ( \hat{a} \bullet \hat{c} ) - \hat{c} ( \hat{a} \bullet \hat{a} ) $
Since $\hat{a} \bullet \hat{a} = 1$, I believe it would make the above:
$\hat{a} ( \hat{a} \bullet \hat{c} ) - \hat{c} ( \hat{a} \bullet \hat{a} ) = \hat{a} ( \hat{a} \bullet \hat{c} ) - \hat{c}$
I don't see how $\hat{a} ( \hat{a} \bullet \hat{c} )=0$ to make the identity work. I must be missing something super basic but for the life of me I can't figure out what it is. Please help.