I landed upon this expression while solving a problem; $$\vec a×\vec b (\vec a . \vec c)-\vec a×\vec c(\vec a .\vec b )$$ To simplify this, I thought of factoring the $\vec a$ out, and it seemed okay to do so, since dot products are distributive. But I don't know what's going to happen to the $\vec c$ and $\vec b$ it was dotted with when it's taken out. Together, each pair of dotted vectors formed a scalar multipier for each vector term, but now that I've taken the $\vec a$ out, I have no idea how to treat the other two.
Is taking the $\vec a $ out wrong here? If so, why?
EDIT: I meant the $\vec a$ in the dot product expression, not the one that's crossed.
Hint:
$$(\vec a \cdot \vec c)\vec b-(\vec a \cdot \vec b) \vec c=\vec a \times (\vec b \times \vec c)$$