Show $(a \times b) \times (a \times c) = ( ( a \times b ) \cdot c ) a$

Question

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

Answer 1

Simply use the vector triple product formula, https://en.m.wikipedia.org/wiki/Triple_product

Let u=a×b Then our question reduces to u×(a×c). Opening up, ((a×b)•c)a - ((a×b)•a)c

The right one becomes 0. And your answer remains.

Answer 2

Hint: the left-hand side's $i$th component is $\epsilon_{ijk}\epsilon_{jlm}a_lb_m\epsilon_{kno}a_nc_o$, while the right-hand side's $i$th component is $\epsilon_{jlm}a_lb_mc_ja_i$. To equate these, use $\epsilon_{ijk}\epsilon_{kno}=\delta_{in}\delta_{jo}-\delta_{io}\delta_{jn}$.