Spin operator in an arbitrary direction

Question

let \(\ \hat{n}=(\sin{\phi}\cos{\theta},\sin{\phi}\cos{\theta},\cos{\phi})\)

How to prove \(\vec{S}\cdot\hat{n}\) is the spin operator in an arbitrary direction rigorously?

Comments

I recommend writing the dot product in the adjoint representation and comparing it to the corresponding 3d rotation matrix.