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