a.) What is the expected magnitude of the sum of two unit vectors in the Cartesian plane if the directions of both vectors are randomly chosen from the interval $[−\pi, \pi]$?
A.) $\dfrac43$ B.) $\sqrt2$ C.) $\dfrac4\pi$ D.) $\dfrac9{2\pi}$
This kind of problem involve the same idea of setting up an integral form of expected value, which would be the integral of $xf(x)$ from $[x_1,x_2]$, only I don't know what $f(x)$ would be. I'm not sure if this is the right mindset to use, but any help would greatly be appreciated! :)
Edit: Got rid of second problem due to divergence issues with the expected integral.
Hints: For a), observe that the problem is unchanged if one of the unit vectors is fixed as $(1, 0)$. Now, can you think of a single parameter $\theta$ to describe the other random unit vector, and describe the magnitude of the sum of the vectors in terms of $\theta$?