This is an applied problem, which arises from the problem of reorienting of a sensor axes according to particle displacement directions:
Consider a sensor which is located inside the solid substance. This sensor is capable of detecting the substance oscillations along each of the three axes (usually orthogonal, but generally, any non-degenerated (non-coplanar) basis). This sensor produces a recording of the detected oscillations, called a trace, containing the displacement sampled at high-frequency, capable to capture any oscillation frequency existing in the substance. There are three traces, one for each axis.
Consider that at some period of time the sensor registers an interference of an event, consisting of the compression wave, and two shear waves. Traces now contain a recordings of the event, effectively, a projections of the oscillations on the sensor's axes.
How do I now virtually re-orient the axes, that is, perform the linear transform of the traces, so that they will be oriented each along a corresponding wave displacement vector, and thus will contain only single wave event recording in each of the traces after the transformation?
EDIT1: Actually, we have a 3D-curve $r(t)$ of sensor motion, which is represented by sensor axial readings $r_1(t), r_2(t), r_3(t)$. The task is to find the "primal" directions of $r(t)$ movements and re-orient axes to those directions $q_1(t), q_2(t), q_3(t)$.