Let be nonzero vectors in . Further, assume , and that is parallel to . Prove that is parallel to if and only if is parallel to .

*Proof.* Assume that is parallel to . Since both and are parallel to , they are parallel to each other by the previous exercise (Section 12.4, Exercise #9). So, we have for some nonzero scalar . This implies

Hence, is parallel to (we know since is a nonzero vector by hypothesis). But then we have and parallel to which implies (again, by the previous exercise) that they are parallel to each other.

Conversely, assume is parallel to . Since and are parallel to we know and for nonzero scalars and . Hence,

We know since is nonzero by hypothesis. Therefore, is parallel to