Zadanie 2e, Paper 3, maj 2025, TZ2, Matematyka AA IB HL

In the spherical model the world has radius 5000 km. Coordinates are in thousands of kilometres, with origin at its centre. The north pole P and station N have vectors $\mathbf p=(0,0,5)$ and $\mathbf n=(0,5,0)$. The positive x direction points west along the equator from N. An equatorial reference point E has vector $\mathbf e=(5,0,0)$. Station M lies north of N on the same meridian, with $\mathbf m=(0,5\cos\theta,5\sin\theta)$ and 0 < θ < π/2. The shorter surface distance NM is 3500 km. Station W is on the equator west of N, with $\mathbf w=(5\sin105^\circ,5\cos105^\circ,0)$. Surface distances follow the shorter great-circle arc.

Find the shortest surface distance between W and M.