Show $a\times (b \times ( c \times a)) = (a\cdot b)(a\times c)$ for vectors $a,b,c$

Question

For $a,b,c\in\Bbb R^3$, show that $$a\times (b \times ( c \times a)) = (a\cdot b)(a\times c)$$

I get to $-a\times(b\cdot a)c$ and eliminating the minus sign gives me $c\times a$ instead of $a\times c$. Can anybody help me to see where I'm going wrong?

Answer 1

From Vector Triple Product you have: $$b \times(c \times a)=c(b \cdot a)-a(b \cdot c)$$ So: $$a \times (b \times(c \times a)) = a \times (c(b \cdot a)-a(b \cdot c))$$ Can you take it from here?