The (time-)"averaged" spin vector is then the projection of the spin onto the direction of :

The primordial vorticity theory predicts that the spin vectors of galaxies are distributed primarily perpendicular to the cluster plane.

The pancake model predicts that the spin vectors of galaxies tend to lie within the cluster plane.

According to the hierarchy model, the directions of the spin vectors should be distributed randomly.

Ferromagnetism occurs when individual atoms are paramagnetic and the spin vectors are aligned parallel to each other in a crystalline material.

As a qualitative concept, the spin vector is often handy because it is easy to picture classically.

The result is that the spin vector undergoes precession, just like a classical gyroscope.

The spin vectors will then start to move in phase and fall back in equilibrium as soon as the radio wave is switched off.

I am not sure how to clearly derive it from the spin vector using vector operations.

There are then detected electrons, the spin vector in the same direction of the momentum vector, with positive helicity.