Past exam of the physics of the Earth as a Planet course of the University of Cambridge / 2016 / Paper 1 / 1 / b / Solution

At latitude $\varphi$ and longitude $\lambda$, $\mathbf{r}=R(\cos \varphi \cos \lambda ,\cos \varphi \sin \lambda ,\sin \varphi )$. Motion is ${\mathbf{v}}_{B/C}={\mathbf{\omega }}_{B/C}\times \mathbf{r}$. A meridian boundary has horizontal normal ${\hat{e}}_{\lambda }=(-\sin \lambda ,\cos \lambda ,0)$, so its across-boundary velocity isFor a transform fault this must vanish along the whole meridian, not only at an isolated point. Thus $5\cos \lambda +12\sin \lambda =0$. Select the solution in the western hemisphere:The other solution is the antipodal meridian in the eastern hemisphere. Geometrically, the transform meridian is in a great-circle plane perpendicular to the Euler angular velocity vector. Its outgoing tangent direction at either pole, expressed in the fixed $x,y$ axes, is ${\hat{t}}_F=(12,-5)/13$.