This is a conceptual question about a problem in Sakurai. I understand how to solve the problem, but there's something about it that irks me, and it feels like I'm missing something.
In the problem, we seek to find the eigenvalues and vectors of matrix S-dot-n in the basis of the z-axis. Where n is defined as an arbitary vector described by $\beta$ and $\alpha$, where $\beta$ is the azumuthal angle and $\alpha$ is radial.
The goal is to find this: $$ \left |\text{s.n;+}\right\rangle = \cos{\frac{\beta}{2}}\left |\text{+}\right\rangle + e^{i \alpha}\sin{\frac{\beta}{2}}\left |\text{-}\right\rangle $$ We start out by constructing this vector, n, in a basis: $$ n = (\sin{\beta}\cos{\alpha}) \hat x + (\sin{\beta}\sin{\alpha}) \hat y + cos{\beta} \hat z $$
And now here's the step that irks me. We represent our each of our bases as their cooresponding Pauli spin matrices and construct our s.n matrix by this. $$ s.n = (\sin{\beta}\cos{\alpha}) \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) + (\sin{\beta}\sin{\alpha}) \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right) + cos{\beta} \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) $$
So I understand the general idea here: We can decompose our n.s into an x, y and z component. And that for each component there is an associated eigenstate basis which can be re-represented in the direction of the z-axis's spin eigenstate basis (as the pauli matrices).
It's hard for me to explain, but something about this step (generating our new operator that finds the spin from our n-vector) really bothers me. Am I missing a step in my logic? Is it possible we can go from n to s.n a litte bit more formally (maybe by appealing to some particular axiom)?