This is based off question 4.30 from Griffith's Introduction to Quanum Mechanics. It asks for the matrix $\textbf{S}_r$ representing the component of spin angular momentum about an axis defined by: $$r = \sin{\theta}\cos{\phi}\hat{i}+\sin\theta\sin\phi\hat{j}+\cos\theta\hat{k}$$
for a spin = $1/2$ particle.The problem is, I can't visualize how the spin vectors relate to spatial coordinates. $$\chi_+ = \begin{pmatrix} 1 \\ 0\end{pmatrix}\ \text{ and } \chi_- = \begin{pmatrix} 0 \\ 1 \end{pmatrix}. $$
So these form a basis of some kind of space, but this is a space I don't understand, for example, why does: $$\chi^{(x)}_+ = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\1 \end{pmatrix} ?$$
I was going to try to use a rotation matrix with angles corresponding to my axis of rotation to try and "rotate" the spin vector onto it, but then I realized that not only do the dimensions not match up, but the angle is π/4, which doesn't make much sense to me. So I guess my question is, how does the 'geometry' of spin work and how can I transform the spins corresponding to a transform I do in space?
Thanks.