Linear algebra. Vectors

Question

Let $ \vec{a}, \vec{b}, \vec{c} $ be non-complanar vectors so that $ \vec{a} \times \vec{b}, \vec{b}, \vec{c} \times \vec{a}$ on the same plane. How can I show that $ \vec{a}$ is perpendicular to $\vec{b}$?

Answer 1

Idea:

We can (probably) write $$\vec{b} = \lambda (\vec{a} \times \vec{b}) + \mu (\vec{c} \times \vec{a})$$ for some scalars $\lambda, \mu$ as the three vectors are co-planar. Now the inner product with $\vec{a}$ of this expression will be $0$, as the inner product is linear and $\vec{a} \cdot (\vec{a} \times \vec{b})=0$ and $\vec{a} \cdot (\vec{c} \times \vec{a}) =0$.

The previous will only fail if $\vec{c} \times \vec{a}$ and $(\vec{a} \times \vec{b})$ are dependent; explore what you can do in that case.