Angle between two vectors

Definition

Let $n$ be a positive integer, and let $u$ and $v$ be nonzero vectors in $\mathbb{R}^n$. Let $u \cdot v = u_1 v_1 + \dots + u_n v_n$ be their dot product, and let $\|u\|$ and $\|v\|$ be their lengths. The angle between $u$ and $v$ is the unique real number $\theta$ with $0 \leq \theta \leq \pi$ such that $$\cos \theta = \frac{u \cdot v}{\|u\| \, \|v\|}.$$ The Cauchy–Schwarz inequality $|u \cdot v| \leq \|u\| \, \|v\|$ makes sure that the quotient is in $[-1, 1]$. Thus $\theta = \arccos\left(\frac{u \cdot v}{\|u\| \, \|v\|}\right)$ exists.