Answer
$12\mathbf{i}-51\mathbf{j}$
Work Step by Step
Addition and subtraction for the given vectors:
$\mathbf{v}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}$ and $\mathbf{w}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j}$
$\begin{align}
& \mathbf{v}+\mathbf{w}=\left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j} \\
& \mathbf{v}-\mathbf{w}=\left( {{a}_{1}}-{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}-{{b}_{2}} \right)\mathbf{j} \\
\end{align}$
Here, ${{a}_{1}}=1,{{a}_{2}}=-2,{{b}_{1}}=-5,{{b}_{2}}=7$
So,
$\begin{align}
& 6\mathbf{v}-3\mathbf{w}=6\left( \mathbf{i}-5\mathbf{j} \right)-3\left( -2\mathbf{i}+7\mathbf{j} \right) \\
& =\left( 6\mathbf{i}-30\mathbf{j} \right)-\left( -6\mathbf{i}+21\mathbf{j} \right) \\
& =\left[ 6-\left( -6 \right) \right]\mathbf{i}+\left( -30-21 \right)\mathbf{j} \\
& =12\mathbf{i}-51\mathbf{j}
\end{align}$
Hence, the vector $6\mathbf{v}-3\mathbf{w}=12\mathbf{i}-51\mathbf{j}$.
