Prove $(\vec a \times \vec a) \times \vec b = \vec a \times (\vec a \times \vec b)$ if and only if $\vec a = \lambda \vec b$

Question

Prove $(\vec a \times \vec a) \times \vec b = \vec a \times (\vec a \times \vec b)$ if and only if $\vec a = \lambda \vec b$. Here's my attempt. Since I need to prove two directions, I prove it one by one. I first want to prove that if $(\vec a \times \vec a) \times \vec b = \vec a \times (\vec a \times \vec b)$, then $\vec a = \lambda \vec b$. Then I write $$\vec a \times \vec a = |\vec a|^2 \sin(\theta)$$. And following the same step, I write $$\vec a \times \vec b = |\vec a||\vec b| \sin(\theta_{2})$$Then I plug it back into my equation and get$$(|\vec a|^2 \sin(\theta)) \times \vec b = \vec a \times (|\vec a||\vec b| \sin(\theta_{2}))$$ However I was stuck on what to do next, since I got a number to cross product with a vector. Could anyone help me with this please? Thanks!

Answer 1

Since $a\times a=0$ for the cross product we then have the condition $$ 0=0\times b=(a\times a)\times b=a\times (a\times b) $$ And this holds iff $a=\lambda b$.