The first postulate of quantum mechanics that can be found in the M. Nielsen and I. Chuang [textbook](https://www.amazon.com/Quantum-Computation-Information-10th-Anniversary/dp/1107002176):

**Postulate 1**: Associated to any isolated physical system is a complex vector space with the inner product (that is, a Hilbert space) known as the state space of the system. The system is completely described by its state vector, which is a unit vector in the system’s state space.

So, If one has some state $|\psi\rangle = \alpha |0\rangle + \alpha |1\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}$, it can be described as a vector in the Hilbert space with $\alpha$ and $\beta$ complex coordinates. Because of normalization $|\alpha|^2 + |\beta|^2 = 1$, the length of that vector is $1$. When we apply a unitary operator the length doesn't change (still $|\alpha'|^2 + |\beta'|^2 = 1$). So unitary operations are just rotations of that vector in the Hilbert space (they don't change the length of it). If we change the basis with a unitary operator then it can be described as a rotation of basis vectors in the Hilbert space (the basis vectors will not change their lengths, they will just point to the other directions).

Not only this. Like the rotations in the real 3D space, the inner product of two vectors is also preserving under unitary transformation ([wiki](https://en.wikipedia.org/wiki/Unitary_operator)). For 3D space, this also implies that the angle between the vectors is not changing. Let's prove the inner product preserving property of the unitary operators:

$$\langle \psi_1 |U^\dagger U|\psi_2\rangle = \langle \psi_1 | I |\psi_2\rangle =  \langle \psi_1 |\psi_2\rangle$$

where we take into account the definition of the unitary operator ($U^\dagger U = I$).