Answer
The obtuse angle $\theta $ is $137.72{}^\circ $.
Work Step by Step
The magnitude of $\mathbf{v}$ is
$\begin{align}
& \left\| \mathbf{v} \right\|=\sqrt{{{3}^{2}}+{{\left( -2 \right)}^{2}}} \\
& =\sqrt{9+4} \\
& =\sqrt{13}
\end{align}$
The magnitude of $\mathbf{w}$ is
$\begin{align}
& \left\| \mathbf{w} \right\|=\sqrt{{{\left( -1 \right)}^{2}}+{{\left( 4 \right)}^{2}}} \\
& =\sqrt{1+16} \\
& =\sqrt{17}
\end{align}$
The value of $\theta $ is
$\begin{align}
& \cos \theta =\frac{3\left( -1 \right)+\left( -2 \right)\left( 4 \right)}{\left\| \mathbf{v} \right\|\left\| \mathbf{w} \right\|} \\
& \cos \theta =\frac{3\left( -1 \right)+\left( -2 \right)\left( 4 \right)}{\sqrt{13}\times \sqrt{17}} \\
& \cos \theta =-0.74 \\
& \theta =137.72{}^\circ
\end{align}$
