Answer
$-\mathbf{i}+2\mathbf{j}$
Work Step by Step
Addition and subtraction for the given vectors:
$\mathbf{v}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}$ and $\mathbf{w}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j}$, we have
$\begin{align}
& \mathbf{v}+\mathbf{w}=\left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j} \\
& \mathbf{v}-\mathbf{w}=\left( {{a}_{1}}-{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}-{{b}_{2}} \right)\mathbf{j} \\
\end{align}$
Here, ${{a}_{1}}=1,{{a}_{2}}=-2,{{b}_{1}}=-5,{{b}_{2}}=7$.
So,
$\begin{align}
& \mathbf{v}+\mathbf{w}=\left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j} \\
& =\left[ 1+\left( -2 \right) \right]\mathbf{i}+\left( -5+7 \right)\mathbf{j} \\
& =-\mathbf{i}+2\mathbf{j}
\end{align}$
Hence, the vector $\mathbf{v}+\mathbf{w}=-\mathbf{i}+2\mathbf{j}$.
