Answer
The vectors $\mathbf{v}$ and $\mathbf{w}$ are orthogonal.
Work Step by Step
The dot product of $\mathbf{v}$ and $\mathbf{w}$ can be obtained as,
$\begin{align}
& \mathbf{v}\cdot \mathbf{w}=\left( 3\mathbf{i}-5\mathbf{j} \right)\cdot \left( 6\mathbf{i}+\frac{18}{5}\mathbf{j} \right) \\
& =3\cdot 6+\left( -5 \right)\cdot \frac{18}{5} \\
& =18-18 \\
& =0
\end{align}$
Since, the dot product of $\mathbf{v}$ and $\mathbf{w}$ is $0$, thus $\mathbf{v}$ and $\mathbf{w}$ are orthogonal vectors.
Hence, $\mathbf{v}$ and $\mathbf{w}$ are orthogonal.
