If $\mathbf{a}\times\mathbf{c}$ is parallel to $\mathbf{b}\times\mathbf{c}$, then $\mathbf{a}$ is parallel to $\mathbf{b}$.

Question

How do we prove the following?

Claim. Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be non-zero vectors. If $\mathbf{a}\times\mathbf{c}$ is parallel to $\mathbf{b}\times\mathbf{c}$, then $\mathbf{a}$ is parallel to $\mathbf{b}$.

(Definition. The vectors $\mathbf{u}$ and $\mathbf{v}$ are parallel if $\mathbf{u}=k\mathbf{v}$ for some $k\neq0$.)

Answer 1

This is not true in general. One counterexample is $a=i,\,b=i+j,\,c=j$.