Answer
The angle between $\mathbf{v}$ and $\mathbf{w}$ is ${{120}^{{}^\circ }}$.
Work Step by Step
Vector $\mathbf{v=}3\cos \frac{5\pi }{3}\mathbf{i}+3\sin \frac{5\pi }{3}\mathbf{j}$ can be written as,
$\begin{align}
& \mathbf{v=}3\cos \frac{5\pi }{3}\mathbf{i}+3\sin \frac{5\pi }{3}\mathbf{j} \\
& =3\left( \frac{1}{2} \right)\mathbf{i}+3\left( -\frac{\sqrt{3}}{2} \right)\mathbf{j} \\
& =\frac{3}{2}\mathbf{i}-\frac{3\sqrt{3}}{2}\mathbf{j}
\end{align}$
And, Vector $\mathbf{w=}2\cos \pi \mathbf{i}+2\sin \pi \mathbf{j}$ can be written as,
$\begin{align}
& \mathbf{w=}2\cos \pi \mathbf{i}+2\sin \pi \mathbf{j} \\
& =2\left( -1 \right)\mathbf{i}+2\left( 0 \right)\mathbf{j} \\
& =-2\mathbf{i}+0\mathbf{j}
\end{align}$
Now, let the angle between $\mathbf{v}$ and $\mathbf{w}$ be $\theta $ such that the angle between the vectors $\mathbf{v}$ and $\mathbf{w}$ can be obtained using the formula $\theta ={{\cos }^{-1}}\left( \frac{\mathbf{v}\cdot \mathbf{w}}{\left| \mathbf{v} \right|\left| \mathbf{w} \right|} \right)$ as,
$\begin{align}
& \theta ={{\cos }^{-1}}\left( \frac{\mathbf{v}\cdot \mathbf{w}}{\left| \mathbf{v} \right|\left| \mathbf{w} \right|} \right) \\
& ={{\cos }^{-1}}\left( \frac{\left( \frac{3}{2}\mathbf{i}-\frac{3\sqrt{3}}{2}\mathbf{j} \right)\cdot \left( -2\mathbf{i}+0\mathbf{j} \right)}{\left( \sqrt{{{\left( \frac{3}{2} \right)}^{2}}+{{\left( \frac{3\sqrt{3}}{2} \right)}^{2}}} \right)\left( \sqrt{{{\left( -2 \right)}^{2}}\mathbf{+}{{\left( 0 \right)}^{2}}} \right)} \right) \\
& ={{\cos }^{-1}}\left( \frac{\left( \frac{3}{2} \right)\cdot \left( -2 \right)+\left( \frac{3\sqrt{3}}{2} \right)\cdot \left( 0 \right)}{\left( \sqrt{9} \right)\left( \sqrt{4} \right)} \right) \\
& ={{\cos }^{-1}}\left( \frac{-3+0}{6} \right)
\end{align}$
Solve ahead to get the result as,
$\begin{align}
& \theta ={{\cos }^{-1}}\left( \frac{-3+0}{6} \right) \\
& ={{\cos }^{-1}}\left( -\frac{1}{2} \right) \\
& ={{120}^{{}^\circ }}
\end{align}$
Hence, the angle, in degrees, between $\mathbf{v}$ and $\mathbf{w}$ is ${{120}^{{}^\circ }}$.
