 Angle between vectors

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.