Answer
$\mathbf{v}\cdot \mathbf{w}=6$ and $\mathbf{v}\cdot \mathbf{v}=10$
Work Step by Step
The dot product, $\mathbf{v}\cdot \mathbf{w}$ as,
$\begin{align}
& \mathbf{v}\cdot \mathbf{w=}\left( 3\mathbf{i+j} \right)\cdot \left( \mathbf{i+}3\mathbf{j} \right) \\
& =3\mathbf{i}\cdot \mathbf{i+}3\mathbf{i}\cdot 3\mathbf{j+j}\cdot \mathbf{i+j}\cdot 3\mathbf{j} \\
& =3\left( 1 \right)+9\left( \mathbf{i}\cdot \mathbf{j} \right)+\left( \mathbf{j}\cdot \mathbf{i} \right)+3\left( \mathbf{j}\cdot \mathbf{j} \right) \\
& =3+9\left( 0 \right)+\left( 0 \right)+3\left( 1 \right)
\end{align}$
Solve ahead to get the result as,
$\begin{align}
& \mathbf{v}\cdot \mathbf{w}=3+9\left( 0 \right)+\left( 0 \right)+3\left( 1 \right) \\
& =3+0+0+3 \\
& =6
\end{align}$
The dot product, $\mathbf{v}\cdot \mathbf{v}$ as,
$\begin{align}
& \mathbf{v}\cdot \mathbf{v=}\left( 3\mathbf{i+j} \right)\cdot \left( 3\mathbf{i+j} \right) \\
& =3\mathbf{i}\cdot 3\mathbf{i+}3\mathbf{i}\cdot \mathbf{j+j}\cdot 3\mathbf{i+j}\cdot \mathbf{j} \\
& =9\left( \mathbf{i}\cdot \mathbf{i} \right)+3\left( \mathbf{i}\cdot \mathbf{j} \right)+3\left( \mathbf{j}\cdot \mathbf{i} \right)+\left( \mathbf{j}\cdot \mathbf{j} \right) \\
& =9\left( 1 \right)+3\left( 0 \right)+3\left( 0 \right)+1
\end{align}$
Solve ahead to get the result as,
$\begin{align}
& \mathbf{v}\cdot \mathbf{v}=9\left( 1 \right)+3\left( 0 \right)+3\left( 0 \right)+1 \\
& =9+0+0+1 \\
& =10
\end{align}$
