A cyclist P is cycling due north at a constant speed of $u$ m/s. Another cyclist Q is due west of P. The speed of Q is constant at $v$ m/s. Find the angle $\theta$ to from east to which $Q$ should travel to minimise the closest distance between $P$ and $Q$
Relative velocity problems in similar settings are very common. I am quite a fan of analytical approaches to mechanics, but when it comes to relative velocity, geometric approaches just seem to crush calculus. This problem can be computed in a single line with geometric considerations. I doubt an analytical approach could do this will, but I'm looking for something that evens the playing field.
My current approach would be to formulate the problem in terms of vectors, where $v_p=(u\cos\theta, u\sin\theta), v_q=(0,v)$ hence we have the relative distance $D$, $D^2=||x_p - x_q||=(ut\cos\theta+d)^2+(ut\sin\theta - vt)^2$. From here we take two partials, one with respect to $t$ to find the $t=T$ at which $D$ is minimum. With this we then minimise with respect to $\theta$ and obtain the answer. This arduous exercise takes no less than fifteen minutes.
I am looking for a good analytical approach to relative velocities in general, but use the given problem as an example to prevent the question being too broad. Most of these problems have the same idea, so I'm hoping there is a quick catch all strategy like the geometers have, rather than only solving this problem.