 Solution

For a Lorentz boost of speed $v$ along $x$, the velocity-addition formula formulas areSubstitute $u_x=u\cos \theta$, $u_y=u\sin \theta$ and take the ratio. The common time-transformation denominator cancels, yieldingTo specify the direction rather than just its tangent, use ${\theta }^{\prime }=atan2(u\sin \theta /{\gamma }_v,u\cos \theta -v)$. This keeps the correct quadrant when the boost reverses the sign of the longitudinal velocity. For physical subluminal speeds the canceled denominator is positive. At $u_x=v$ and nonzero $u_y$, the transformed direction is perpendicular to the boost; if the particle is exactly comoving, its transformed velocity vanishes and the angle is undefined.