Answer
$$\left\langle\frac{1}{2},-\frac{\sqrt{3}}{2}, 0\right\rangle$$
$$\left\langle-\frac{\sqrt{3}}{2},-\frac{1}{2}, 9\right\rangle$$
Work Step by Step
Given $$\mathbf{r}(\theta)=\langle\sin \theta, \cos \theta, \cos 3 \theta\rangle, \quad \theta=\frac{\pi}{3}$$
Velocity is given by
\begin{aligned}
\mathbf{v}(\theta) &=\mathbf{r}^{\prime}(\theta)\\
&=\langle\cos \theta,-\sin \theta,- 3 \sin 3 \theta\rangle \\
\mathbf{v}\left(\frac{\pi}{3}\right) &=\left\langle\cos \frac{\pi}{3},-\sin \frac{\pi}{3},-3 \sin \pi\right\rangle\\
&=\left\langle\frac{1}{2},-\frac{\sqrt{3}}{2}, 0\right\rangle
\end{aligned}
and acceleration is given by
\begin{aligned}
\mathbf{a}(\theta) &=\mathbf{v}^{ \prime}(\theta)\\
&=\langle-\sin \theta,-\cos \theta,- 9 \cos 3 \theta\rangle \\
\mathbf{a}\left(\frac{\pi}{3}\right)&=\left\langle-\sin \frac{\pi}{3},-\cos \frac{\pi}{3},-9 \cos \pi\right\rangle\\
&=\left\langle-\frac{\sqrt{3}}{2},-\frac{1}{2}, 9\right\rangle
\end{aligned}
