Answer
$\textbf{v} = \left<2, 2\sqrt{3} \right>$.
Work Step by Step
We can consider $\textbf{v}$ the hypotenuse of a right triangle with $\theta = \frac{\pi}{3}$ adjacent to the horizontal leg $v_x$ (which is the x-component) and opposite the vertical leg $v_y$ (which is the y-component). Since $|\textbf{v}| = 4$ is the length of $\textbf{v}$, we can use trigonometry to find the legs:
$v_x = |\textbf{v}| \cos \theta = 4\cos \frac{\pi}{3} = 2$
$v_y = |\textbf{v}| \sin \theta = 4\sin \frac{\pi}{3} = 2\sqrt{3}$
Therefore, written in component form, $\textbf{v} = \left<2, 2\sqrt{3} \right>$.
