Answer
The angle between $\mathbf{v}$ and $\mathbf{w}$ is $\theta \approx 138{}^\circ $.
Work Step by Step
We have
$\begin{align}
& \cos \theta =\frac{\mathbf{v}\cdot \mathbf{w}}{\left\| \mathbf{v} \right\|\left\| \mathbf{w} \right\|} \\
& =\frac{\left( -5\mathbf{i}+2\mathbf{j} \right)\cdot \left( 2\mathbf{i}-4\mathbf{j} \right)}{\left( \sqrt{{{\left( -5 \right)}^{2}}+{{2}^{2}}} \right)\cdot \left( \sqrt{{{2}^{2}}+{{\left( -4 \right)}^{2}}} \right)} \\
& =\frac{\left( -5\left( 2 \right) \right)+2\left( -4 \right)}{\left( \sqrt{29} \right)\cdot \left( \sqrt{20} \right)} \\
& =-\frac{18}{\left( \sqrt{580} \right)}
\end{align}$
So,
$\begin{align}
& \theta ={{\cos }^{-1}}\left( -\frac{18}{\left( \sqrt{580} \right)} \right) \\
& \approx 138{}^\circ
\end{align}$
Therefore, the angle between $\mathbf{v}$ and $\mathbf{w}$ is $\theta \approx 138{}^\circ $.
