PDF of an angle between two 2d random vectors

Question

I have the following problem I'd like to solve. I am drawing vectors in 2d from Gaussian distributions. The first vector $\rho_1$ is drawn from $N(\mu_1, \Sigma_1)$ and the second, $\rho_2$ from $N(\mu_2, \Sigma_2)$. In both cases $\rho$ and $\mu$ in are 2d vectors and $\Sigma$ is a $2\times2$ matrix. One can assume that one of these vectors is lying on the $x$-axis without any loss of generality I believe. Is it possible to compute the PDF of the angle between $\rho_1$ and $\rho_2$? If it makes it any easier, the simplification $\Sigma=\sigma^2I_2$, where $I_2$ is the $2\times2$ identity matrix, is acceptable.