Well, I hope you at least know the bases or axioms of vector space. In particular every vector space has an "additive identity"or "0 vector" (typically represented by "0" but not necessarily having anything to do with the number 0) such that, for any vector, v, v+ 0= v.
Here you are defining addition by \(\displaystyle \begin{pmatrix} u_0 \\ u_1\\u_2\end{pmatrix}+ \begin{pmatrix}v_0 \\ v_1 \\ v_2 \end{pmatrix}= \begin{pmatrix}u_0+ v_0+ 4 \\ u_1+v_1+ 2 \\ u_2+ v_2+ 3\end{pmatrix}\) so writing the additive identity (0 vector) as \(\displaystyle \begin{pmatrix} a \\ b \\ c \end{pmatrix}\) we must have\(\displaystyle \begin{pmatrix} u_0 \\ u_1\\u_2\end{pmatrix}+ \begin{pmatrix}a \\ b \\ c \end{pmatrix}= \begin{pmatrix}u_0+ a+ 4 \\ u_1+b+ 2 \\ u_2+ c+ 3\end{pmatrix}= \begin{pmatrix} u_0 \\ u_1\\u_2\end{pmatrix}\).
So we must have \(\displaystyle u_0+ a+ 4= u_0\), \(\displaystyle u_1+ b+ 2= u_1\), and \(\displaystyle u_2+ b+ 3= u_2\) so a= -4, b= -2, and c= -3. That is, the additive identity, or 0 vector, is \(\displaystyle \begin{pmatrix}a \\ b \\ c\end{pmatrix}= \begin{pmatrix}-4\\ -2 \\ -3 \end{pmatrix}\).