Answer
See the proof below.
Work Step by Step
Consider the following statement:
In a vector space the zero vector is unique.
We will prove by contradiction that the statement is true.
$\textbf{Proof:}$ Let $V$ be a vector space and suppose there exist two distinct vectors $\textbf{0}$ and $\textbf{0}'$ such that
$u+\textbf{0}=u$ and $u+\textbf{0}'=u$ for all $u\in V$.
Thus, by taking $u=\textbf{0}'$ in the first equation and $u=\textbf{0}$ in the second equation above we obtain that
$$\textbf{0}'+\textbf{0}=\textbf{0}'$$
and
$$\textbf{0}+\textbf{0}'=\textbf{0}.$$
Now, by the commutativity of the addition in the vector space $V$, we have that $\textbf{0}+\textbf{0}'=\textbf{0}'+\textbf{0}$ and then, it follows from the two equalities above that $\textbf{0}=\textbf{0}'$ which contradicts our assumption that $\textbf{0}\neq\textbf{0}'$.
Hence, the statement is true.
