# An explanation for the form of the spin component operator for a measurement made at an angle in the $x$-$y$ plane

I have three Pauli spin matrices corresponding to the spin states along each Cartesian axis. For some rotation about $$\Phi_n$$ about the $$z$$-axis we have:$$S_n=S_xsin(\Theta)cos(\Phi) +S_ysin(\Theta)sin(\Phi)+S_zcos(\Theta)$$ Where $$\Theta$$ is a rotation about the $$z$$-axis and $$\Phi$$ is a rotation about the $$z$$-axis in the $$x$$-$$y$$ plane. I don't understand how we go from this to the matrix representation: $${S_n=\frac{\hbar}{2} \begin{pmatrix} cos(\Theta) & sin(\Theta)e^{-i\Phi}\\ sin(\Theta)e^{i\Phi} & -cos(\Theta)\\ \end{pmatrix}}$$ Please can someone explain the origin of the complex exponential terms?

It's just Euler's $$\cos \Phi+i\sin \Phi= e^{i\Phi}$$ compbined with the usual forms of Pauli's $$\sigma_x$$ and $$\sigma_y$$.