Answer
$\mathbf{v}$ and $\mathbf{w}$ are not orthogonal.
Work Step by Step
Dot product of $\mathbf{v}$ and $\mathbf{w}$ can be obtained as,
$\begin{align}
& \mathbf{v}\cdot \mathbf{w}=\left( 5\mathbf{i}-5\mathbf{j} \right)\cdot \left( \mathbf{i}-\mathbf{j} \right) \\
& =5\cdot 1+\left( -5 \right)\cdot \left( -1 \right) \\
& =5+5 \\
& =10
\end{align}$
As the dot product of $\mathbf{v}$ and $\mathbf{w}$ is not zero, thus, $\mathbf{v}$ and $\mathbf{w}$ are not orthogonal.
Hence, $\mathbf{v}$ and $\mathbf{w}$ are not orthogonal.
