Answer
$4$
Work Step by Step
Addition and subtraction in vectors:
For the two given vectors $\mathbf{v}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}$ and $\mathbf{w}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j},$
$\mathbf{v}+\mathbf{w}=\left( {{a}_{1}}+{{a}_{2}} \right)\mathbf{i}+\left( {{b}_{1}}+{{b}_{2}} \right)\mathbf{j}$
Dot product of vectors:
For the two given vectors $\mathbf{v}={{a}_{1}}\mathbf{i}+{{b}_{1}}\mathbf{j}$ and $\mathbf{w}={{a}_{2}}\mathbf{i}+{{b}_{2}}\mathbf{j},$
$\mathbf{v}\cdot \mathbf{w}={{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}$ and
$\cos \theta =\frac{\left\| \mathbf{v} \right\|\cdot \left\| \mathbf{w} \right\|}{\mathbf{v}\cdot \mathbf{w}}$
where $\theta $ is the angle between the vectors $\mathbf{v}$ and $\mathbf{w}$.
So,
$\begin{align}
& \mathbf{v}+\mathbf{w}=\left( 1+3 \right)\mathbf{i}+\left( -1-7 \right)\mathbf{j} \\
& =4\mathbf{i}-8\mathbf{j}
\end{align}$
Thus,
$\begin{align}
& \mathbf{u}\cdot \left( \mathbf{v}+\mathbf{w} \right)=\left( 5\mathbf{i}+2\mathbf{j} \right)\cdot \left( 4\mathbf{i}-8\mathbf{j} \right) \\
& =5\left( 4 \right)+2\left( -8 \right) \\
& =20-16 \\
& =4
\end{align}$
