How to justify that a spin operator in an arbitrary direction is given by
$$S=\sum_i \sigma_in_i$$
Where $\sigma_i$ are Pauli matrices and $n_i$ is a component of an arbitrary unit vector in $i'th$ direction.
I deliberately did not write the above expression as $S=\vec\sigma\cdot\vec n$ because i am not sure how one can treat spin operator as a vector in this case. I feel like this sought of notation would make sense if one studies operator transformations but i am not aquanted with those yet.
