Prove some properties of parallel, nonzero vectors

Let $A,B,C,D$ be nonzero vectors in $\mathbb{R}^n$ . Further, assume $C = A + B$ , and that $A$ is parallel to $D$ . Prove that $C$ is parallel to $D$ if and only if $B$ is parallel to $D$ .

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

$C = xC + B \quad \implies \quad B = (1-x)C.$

Hence, $B$ is parallel to $C$ (we know $1-x \neq 0$ since $B$ is a nonzero vector by hypothesis). But then we have $B$ and $D$ parallel to $C$ which implies (again, by the previous exercise) that they are parallel to each other.

Conversely, assume $B$ is parallel to $D$ . Since $A$ and $B$ are parallel to $D$ we know $A = xD$ and $B = yD$ for nonzero scalars $x$ and $y$ . Hence,