$ [(\vec{a}+\vec{b})\times(\vec{b}+\vec{c})]\cdot(\vec{c}+\vec{a})=2\vec{c}\cdot(\vec{b}\times\vec{a})$
I'm supposed to prove that this is true for all vectors $a,b,c$, but I keep getting $4(\vec{a}\times \vec{b})\cdot \vec{c}=0$, which is obviously not true.
Seems to me like there is a mistake in the problem, that it should be $\vec{a}\times\vec{b}$ instead of $\vec{b}\times\vec{a}$. Or I'm doing it wrong?
I'm using facts that $\vec{a}\times\vec{a}=0$ and $[\vec{a},\vec{b},\vec{c}]=[\vec{c},\vec{b},\vec{a}]=-[\vec{b},\vec{a},\vec{c}]$.