Using concepts of geometric orthogonality and linear independence, we logically deduce the form of
the Pauli spin matrices and the relationships between the three spatially orthogonal basis sets of
the spin-1/2 system. Rather than a mathematically rigorous derivation, the relationships are found
by forcing expectation values of the different basis states to have the properties we expect of a
classical, geometric coordinate system. The process highlights the correspondence of quantum angular
momentum with classical notions of geometric orthogonality, even for the inherently non-classical
spin-1/2 system. In the process, differences in and connections between geometrical space and
Hilbert space are illustrated.