Proof if $a \vec v = 0$ then $a = 0$ or $\vec v = 0$

Let $a\in F,\vec v\in V$ and suppose $a\vec v=0$. If $a\neq 0$, then $a^{-1}\in F$ so $$\vec v=(a^{-1}a)\vec v=a^{-1}(a\vec v)=a^{-1}\vec 0=\vec 0$$ thus either $a=0$ or $\vec v=0$.

If you want to prove that "P or Q" holds, it is often useful to assume that one of the conditions fails, from which it may follow readily that the other must hold (which proves the statement).

In this case, we know that non-0 field elements have multiplicative inverses, and can show easily that scalar multiples of the zero vector will again be the zero vector. Then if we assume $a\neq 0$, it readily follows that $v=0$.