Answer
At least one of the vectors $\vec{a}, \vec{b}$ must be the zero vector.
Work Step by Step
Consider some real number $a, b$. Then $a+b = a-b$ is only true when at least one number equals to 0. Similarly, in a vector space, the equality holds only if $\vec{a}$, $\vec{b}$, or both is the additive identity (i.e.) the zero vector.
