Answer
$\mathbf{v}\cdot \mathbf{w}=-6$ and $\mathbf{v}\cdot \mathbf{v}=41$
Work Step by Step
The dot product, $\mathbf{v}\cdot \mathbf{w}$ as,
$\begin{align}
& \mathbf{v}\cdot \mathbf{w=}\left( 5\mathbf{i}-4\mathbf{j} \right)\cdot \left( -2\mathbf{i}-\mathbf{j} \right) \\
& =5\mathbf{i}\cdot \left( -2 \right)\mathbf{i+}5\mathbf{i}\cdot \left( -1 \right)\mathbf{j+}\left( -4 \right)\mathbf{j}\cdot \left( -2 \right)\mathbf{i+}\left( -4 \right)\mathbf{j}\cdot \left( -1 \right)\mathbf{j} \\
& =-10\left( \mathbf{i}\cdot \mathbf{i} \right)-5\left( \mathbf{i}\cdot \mathbf{j} \right)+8\left( \mathbf{j}\cdot \mathbf{i} \right)+4\left( \mathbf{j}\cdot \mathbf{j} \right) \\
& =-10\left( 1 \right)-5\left( 0 \right)+8\left( 0 \right)+4\left( 1 \right)
\end{align}$
Solve ahead to get the result as,
$\begin{align}
& \mathbf{v}\cdot \mathbf{w}=-10\left( 1 \right)-5\left( 0 \right)+8\left( 0 \right)+4\left( 1 \right) \\
& =-10+0+0+4 \\
& =-6
\end{align}$
The dot product, $\mathbf{v}\cdot \mathbf{v}$ as,
$\begin{align}
& \mathbf{v}\cdot \mathbf{v=}\left( 5\mathbf{i}-4\mathbf{j} \right)\cdot \left( 5\mathbf{i}-4\mathbf{j} \right) \\
& =5\mathbf{i}\cdot 5\mathbf{i+}5\mathbf{i}\cdot \left( -4 \right)\mathbf{j+}\left( -4 \right)\mathbf{j}\cdot 5\mathbf{i+}\left( -4 \right)\mathbf{j}\cdot \left( -4 \right)\mathbf{j} \\
& =25\left( \mathbf{i}\cdot \mathbf{i} \right)-20\left( \mathbf{i}\cdot \mathbf{j} \right)-20\left( \mathbf{j}\cdot \mathbf{i} \right)+16\left( \mathbf{j}\cdot \mathbf{j} \right) \\
& =25\left( 1 \right)-20\left( 0 \right)-20\left( 0 \right)+16\left( 1 \right)
\end{align}$
Solve ahead to get the result as,
$\begin{align}
& \mathbf{v}\cdot \mathbf{v}=25\left( 1 \right)-20\left( 0 \right)-20\left( 0 \right)+16\left( 1 \right) \\
& =25+0+0+16 \\
& =41
\end{align}$
