Answer
If $\mathbf{v}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}$, and $\mathbf{w}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j}$, then
$\mathbf{v}+\mathbf{w}=\left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j}$
$\mathbf{v}-\mathbf{w}=\left( {{a}_{1}}-{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}-{{b}_{2}} \right)\mathbf{j}$
$k\mathbf{v}=k{{a}_{1}}\mathbf{i}+k{{a}_{2}}\mathbf{j}$
Work Step by Step
The values of the above expression can be calculated by substituting the values of $\mathbf{v}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}$, and $\mathbf{w}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j}$ as below:
$\begin{align}
& \mathbf{v}+\mathbf{w}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}+{{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j} \\
& =\left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j}
\end{align}$
$\begin{align}
& \mathbf{v}-\mathbf{w}=\left( {{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j} \right)-\left( {{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j} \right) \\
& =\left( {{a}_{1}}-{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}-{{b}_{2}} \right)\mathbf{j}
\end{align}$
And,
$\begin{align}
& k\mathbf{v}=k\left( {{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j} \right) \\
& =k{{a}_{1}}\mathbf{i}+k{{a}_{2}}\mathbf{j}
\end{align}$
