If $v$, $w$, and $v + w$ are nonzero, and $v$ and $v + w$ parallel, then $v$ and $w$ are parallel.

Question

I am quite sure this is true but I do not know where to start.

If $v$, $w$, and $v + w$ are nonzero, and $v$ and $v + w$ parallel, then $v$ and $w$ are parallel.

Answer 1

Hint: Being parallel means one is a scalar multiple of the other. Start with $v=k(v+w)$ and work from there

Answer 2

Since we know v and v+w are parallel, we have

v = a(v+w)

v = av + aw

v - av = aw

(1-a)v = aw

Since a is a scalar, subtracting it from 1 still results in a scalar. From this, we can see that v and w are scalar multiples of each other and are therefore parallel.