Quantum State Vectors
An extended comparison of quantum state 'vectors', and the regular ol' physics vectors.
Vectors
The state of a quantum system--all we can know about it--is represented by a "ket", written $$\ket{\psi}.\nonumber$$ That's the greek letter $\psi$, spelled as "psi" and pronounced as sigh.
Many physics quantities can be represented by vectors. $$\myv A\nonumber.$$
Completeness
Any possible quantum state (completeness) can be described as a linear superposition of basis states. The basis states of the $Z$ analyzer are $\ket +$, (spin up, or $S_z=+\hbar/2$) and $\ket -$ (spin down, or $S_z=-\hbar/2$). So any $\ket \psi$ of an electron (a spin-1/2 particle) can be written as $$\ket \psi = a\ket + + b\ket -.$$ where $a$ and $b$ are complex numbers.
There are only 2 possible outcomes for a measurement of of the $z$ component of the spin of an electron. But there are other fundamental particles that have 3 or more possible $S_z$ components, and this scheme can readily be generalized to describe state vectors in any old $n$-dimensional space.
Any vector (completeness) can be written as a linear sum of unit vectors in a particular coordinate system: $$\myv A=A_x\uv i+A_y\uv j,$$ where $A_x$ and $A_y$ are real numbers.
In addition to 2-d vectors, we routinely use 3-d vectors to describe positions, velocities, etc, in 3-d space. In special relativity we can talk about 4-d space-time vectors. And in principle this scheme can be generalized to any desired number of dimensions.