The Dot Product in Euclidean Space

Math 303: Section 27

Dr. Janssen

Intro

\[ \def\R{{\mathbb R}} \def\P{{\mathbb P}} \def\B{{\mathcal B}} \def\C{{\mathcal C}} \def\S{{\mathcal S}} \def\b{{\mathbf{b}}} \def\c{{\mathbf{c}}} \def\x{{\mathbf{x}}} \def\y{{\mathbf{y}}} \def\u{{\mathbf{u}}} \def\v{{\mathbf{v}}} \def\w{{\mathbf{w}}} \def\z{{\mathbf{z}}} \def\M{{\mathcal{M}}} \DeclareMathOperator{\null}{Nul} \DeclareMathOperator{\span}{Span} \DeclareMathOperator{\dim}{dim} \newcommand{\set}[1]{\left\{ {#1} \right\}} \newcommand{\setof}[2]{{\left\{#1\,\colon\,#2\right\}}} \newcommand{\norm}{1}{{\left|\! \left| #1 \right| \! \right|}} \]

Recall

The length/magnitude/norm of the vector \(\v = \left[\begin{matrix} v_1 \\ v_2 \end{matrix}\right]\) in \(\R^2\) is

\[ \norm{\v} = \sqrt{v_1^2 + v_2^2}. \]

The dot product of vectors \(\u = [ u_1 \ u_2 \ \cdots \ u_n]^\textsf{T}\) and \(\v = [ v_1 \ v_2 \ \cdots \ v_n]^\textsf{T}\) in \(\R^n\) is

\[ \u\cdot \v = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n. \]