Answer
.
Work Step by Step
Vectors are $u={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j},v={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j},$ and $w={{a}_{3}}\mathbf{i}+{{b}_{3}}\mathbf{j}$. Now, put these vectors in the expression $u+v=v+u$. Then,
$\begin{align}
& \left( {{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j} \right)+\left( {{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j} \right)=\left( {{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j} \right)+\left( {{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j} \right) \\
& \left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j}=\left( {{a}_{2}}+{{a}_{1}} \right)\mathbf{i}+\left( {{b}_{2}}+{{b}_{1}} \right)\mathbf{j} \\
& \left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j}=\left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j} \\
\end{align}$
Left side and right side are equal. Hence, $u+v=v+u$.
