Angle between vectors ($\cos \theta =\mathbf{a}\cdot \mathbf{b}/(\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert )$)

For nonzero real vectors in a Euclidean space, their angle $\theta \in [0,\pi ]$ is determined by the displayed dot product formula. The Cauchy-Schwarz inequality puts the quotient in $[-1,1]$, so the inverse cosine is defined. Perpendicular vectors have angle $\pi /2$, parallel vectors pointing the same way have angle zero, and those pointing in opposite directions have angle $\pi$. A zero vector has no uniquely defined direction and hence no angle by this definition.

For nonzero vectors in a complex inner product space, the displayed quotient defines the ordinary real angle in the underlying real inner product space, whose inner product is $\mathrm{Re}⟨u,v⟩$. The complex Cauchy-Schwarz inequality bounds it between $-1$ and $1$. This convention depends on the vectors' phase factors. The different phase-invariant convention $|⟨u,v⟩|/(\Vert u\Vert \Vert v\Vert )$ measures an angle between complex one-dimensional subspaces; the two conventions should not be interchanged.