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*}