I have got nowhere with this question: Non-zero non-parallel vectors a, b and c are such that b x c = c x a. Prove that a + b = kc for some scalar k.
By the given we know that $\displaystyle \vec{c}\times(\vec{b}\times \vec{c})=\vec{c}\times(\vec{c}\times \vec{a})$
But $\displaystyle \vec{c}\times(\vec{b}\times \vec{c})=(\vec{c}\cdot\vec{c})\vec{b}-(\vec{c}\cdot\vec{b})\vec{c}$
Now expand the RHS and solve for $\displaystyle \vec{a}+\vec{b}$.