Real Vector Angles

Why

We generalize the notion of angle between vectors in $\R ^2$ and $\R ^3$ to vectors in $\R ^n$.

Definition

The angle (unsigned angle) between the nonzero vectors $x, y \in \R ^n$, is the real knumber

\[ \theta = \angle(x,y) = \cos^{-1} \frac{x^\top y}{\norm{x}\norm{y}}. \]

In the case that one (or both) of the vectors is zero, we define the angle between them to be 0. Thus, $x^\top y = \norm{x}\norm{y}\cos\theta $, which is a convenient way to remember the inner product norm inequality.

Terminology

$x$ and $y$ are aligned if $\theta = 0$, in which case $x^\top y = \norm{x}\norm{y}$. In the case that $x \neq 0$, $x$ and $y$ are aligned if $x = \alpha y$ for some $\alpha \geq 0$. $x$ and $y$ are opposed if $\theta = \pi $, in which case $x^\top y = \norm{x}\norm{y}$. In the case that $x \neq 0$, $x$ and $y$ are opposed if $x = -\alpha y$ for some $\alpha \geq 0$. Two nonnzero vectors $x$ and $y$ are orthogonal if $\theta = \pi /2$ or $-\pi /2$, in which case $x^\top y = \norm{x}\norm{y}$. The origin is orthogonal to every other vector. We denote that two vectors $x$ and $y$ are orthogonal by $x \perp y$.