For a given quantum state, one could think of a spin vector ⟨ ⟨ S ⟩ ⟩ whose components are the expectation values of the spin components along each axis, i.e., ⟨ ⟨ S ⟩ ⟩ = [ ⟨ ⟨ S x ⟩ ⟩ , ⟨ ⟨ S y ⟩ ⟩ , ⟨ ⟨ S z ⟩ ⟩ ] . This vector then would describe the "direction" in which the spin is pointing, corresponding to the classical concept of the axis of rotation. It turns out that the spin vector is not very useful in actual quantum-mechanical calculations, because it cannot be measured directly: s x, s y and s z cannot possess simultaneous definite values, because of a quantum uncertainty relation between them. However, for statistically large collections of particles that have been placed in the same pure quantum state, such as through the use of a Stern–Gerlach apparatus, the spin vector does have a well-defined experimental meaning: It specifies the direction in ordinary space in which a subsequent detector must be oriented in order to achieve the maximum possible probability (100%) of detecting every particle in the collection. For spin- ⁠ 1 / 2 ⁠ particles, this probability drops off smoothly as the angle between the spin vector and the detector increases, until at an angle of 180°—that is, for detectors oriented in the opposite direction to the spin vector—the expectation of detecting particles from the collection reaches a minimum of 0%.