Discover Linear Algebra

Section 13.2 Terminology and notation

norm (of a vector $\uvec{v}$)

the quantity $\unorm{v} = \sqrt{v_1^2+v_2^2+\dotsb+v_n^2}\text{;}$ also called the length or magnitude of $\uvec{v}$

unit vector

a vector whose norm is equal to $1$

normalization (of a vector $\uvec{v}$)

the unit vector $\displaystyle \frac{1}{\unorm{v}}\,\uvec{v}$

distance (between two vectors $\uvec{u}$ and $\uvec{v}$)

the distance between the terminal points of the two vectors when their initial points are placed at the same point; can be computed as $\norm{\uvec{u}-\uvec{v}}$ (or equivalently as $\norm{\uvec{v}-\uvec{u}}$)

dot product (of two vectors $\uvec{u}$ and $\uvec{v}$ of the same dimension)

the quantity

$$\begin{equation*} \udotprod{u}{v} = u_1 v_1 + u_2 v_2 + \dotsb + u_n v_n \text{;} \end{equation*}$$

also referred to as the Euclidean inner product or standard inner product of $\uvec{u}$ and $\uvec{v}$

angle (between two vectors $\uvec{u}$ and $\uvec{v}$ of the same dimension)

the angle $\theta$ satisfying both

$$\begin{align*} 0 \amp\le \theta \le \pi \amp \amp\text{and} \amp \cos\theta \amp= \frac{\udotprod{u}{v}}{\unorm{u} \unorm{v}} \end{align*}$$