Answer
$\mathbf{v}\cdot \mathbf{w}=-19$ and $\mathbf{v}\cdot \mathbf{v}=53$
Work Step by Step
The dot product, $\mathbf{v}\cdot \mathbf{w}$ as,
$\begin{align}
& \mathbf{v}\cdot \mathbf{w=}\left( 7\mathbf{i}-2\mathbf{j} \right)\cdot \left( -3\mathbf{i}-\mathbf{j} \right) \\
& =7\mathbf{i}\cdot \left( -3 \right)\mathbf{i+}7\mathbf{i}\cdot \left( -1 \right)\mathbf{j+}\left( -2 \right)\mathbf{j}\cdot \left( -3 \right)\mathbf{i+}\left( -2 \right)\mathbf{j}\cdot \left( -1 \right)\mathbf{j} \\
& =-21\left( \mathbf{i}\cdot \mathbf{i} \right)-7\left( \mathbf{i}\cdot \mathbf{j} \right)+6\left( \mathbf{j}\cdot \mathbf{i} \right)+2\left( \mathbf{j}\cdot \mathbf{j} \right) \\
& =-21\left( 1 \right)-7\left( 0 \right)+6\left( 0 \right)+2\left( 1 \right)
\end{align}$
Solve ahead to get the result as,
$\begin{align}
& \mathbf{v}\cdot \mathbf{w}=-21\left( 1 \right)-7\left( 0 \right)+6\left( 0 \right)+2\left( 1 \right) \\
& =-21+0+0+2 \\
& =-19
\end{align}$
The dot product, $\mathbf{v}\cdot \mathbf{v}$ as,
$\begin{align}
& \mathbf{v}\cdot \mathbf{v=}\left( 7\mathbf{i}-2\mathbf{j} \right)\cdot \left( 7\mathbf{i}-2\mathbf{j} \right) \\
& =7\mathbf{i}\cdot 7\mathbf{i+}7\mathbf{i}\cdot \left( -2 \right)\mathbf{j+}\left( -2 \right)\mathbf{j}\cdot 7\mathbf{i+}\left( -2 \right)\mathbf{j}\cdot \left( -2 \right)\mathbf{j} \\
& =49\left( \mathbf{i}\cdot \mathbf{i} \right)-14\left( \mathbf{i}\cdot \mathbf{j} \right)-14\left( \mathbf{j}\cdot \mathbf{i} \right)+4\left( \mathbf{j}\cdot \mathbf{j} \right) \\
& =49\left( 1 \right)-14\left( 0 \right)-14\left( 0 \right)+4\left( 1 \right)
\end{align}$
Solve ahead to get the result as,
$\begin{align}
& \mathbf{v}\cdot \mathbf{v}=49\left( 1 \right)-14\left( 0 \right)-14\left( 0 \right)+4\left( 1 \right) \\
& =49+0+0+4 \\
& =53
\end{align}$
