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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Dot Product">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\begin{document}


\section{The Dot Product\label{Dot product}}

If $\mathbf{a}=\left\langle a_{1},a_{2},a_{3}\right\rangle $ and $\mathbf{b}%
=\left\langle b_{1},b_{2},b_{3}\right\rangle $ are vectors in $V_{3}$ then
the \textbf{dot product} of $\mathbf{a}$ and $\mathbf{b}$ is the number $%
\mathbf{a}\cdot \mathbf{b}$ given by 
\[
\mathbf{a}\cdot \mathbf{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3} 
\]
The dot product of two vectors is a scalar, not a vector. The dot product is
also called the \textbf{scalar product} or \textbf{inner product}. The dot
product in $V_{2}$ is defined similarly. In general, if $n>0$ and $\mathbf{a}%
=\left\langle a_{1},a_{2},\ldots ,a_{n}\right\rangle $ and $\mathbf{b}%
=\left\langle b_{1},b_{2},\ldots ,b_{n}\right\rangle $ are vectors in $V_{n}$%
, then the dot product of $\mathbf{a}$ and $\mathbf{b}$ is given by 
\[
\mathbf{a}\cdot \mathbf{b}=\sum\limits_{m=1}^{n}a_{m}b_{m} 
\]

\subsubsection{Properties of the Dot Product}

Suppose $\mathbf{a},\mathbf{b,c}$ are vectors in $V_{3}$ and $c$ is a
scalar. Then 
\begin{eqnarray*}
\mathbf{a}\cdot \mathbf{a} &=&\left| \mathbf{a}\right| ^{2}\text{ or }\sqrt{%
\mathbf{a}\cdot \mathbf{a}}=\left| \mathbf{a}\right| \\
\mathbf{a}\cdot \mathbf{b} &=&\mathbf{b}\cdot \mathbf{a} \\
\mathbf{a}\cdot \left( \mathbf{b}+\mathbf{c}\right) &=&\mathbf{a}\cdot 
\mathbf{b+a}\cdot \mathbf{c} \\
c\mathbf{a}\cdot \mathbf{b} &=&c\left( \mathbf{a}\cdot \mathbf{b}\right) =%
\mathbf{a}\cdot c\mathbf{b} \\
\mathbf{0}\cdot \mathbf{a} &=&0
\end{eqnarray*}

In $V_{3}$, the dot product has a interesting geometric interpretation. If $%
\mathbf{a},\mathbf{b}$ are vectors in $V_{3}$, then the \textbf{angle }$%
\theta $ \textbf{between }$\mathbf{a}$ \textbf{and }$\mathbf{b}$ is defined
to be the angle between the representations of $\mathbf{a}$ and $\mathbf{b}$
that start at the origin, where $0\leq \theta \leq \pi .$ If $\mathbf{a}$
and $\mathbf{b}$ are parallel vectors, then $\theta =0$ or $\theta =\pi $.

\begin{theorem}
If $\theta $ is the angle between two vectors $\mathbf{a}$ and $\mathbf{b}$%
\textbf{, }then 
\[
\mathbf{a}\cdot \mathbf{b}=\left| \mathbf{a}\right| \left| \mathbf{b}\right|
\cos \theta 
\]
If $\mathbf{a}$ and $\mathbf{b}$ are \emph{nonzero }vectors, then this
result may be rewritten as 
\[
\cos \theta =\frac{\mathbf{a}\cdot \mathbf{b}}{\left| \mathbf{a}\right|
\left| \mathbf{b}\right| } 
\]
\end{theorem}

Two \emph{nonzero }vectors $\mathbf{a}$ and $\mathbf{b}$ are called \textbf{%
perpendicular }or \textbf{orthogonal }if $\mathbf{a}\cdot \mathbf{b}=0$.

\begin{theorem}
Given two \emph{nonzero }vectors $\mathbf{a}$ and $\mathbf{b}$, with $\theta 
$ the angle between them, then $\mathbf{a}$ and $\mathbf{b}$ are orthogonal
if and only if $\theta =\dfrac{\pi }{2}$.
\end{theorem}

Suppose $\mathbf{a}$ is a \emph{nonzero }vector then the angles $\alpha $, $%
\beta $, and $\gamma $ between $\mathbf{a}$ and the vectors $\mathbf{i}$%
\textbf{, }$\mathbf{j}$\textbf{, and }$\mathbf{k}$ respectively are called
the \textbf{direction angles }of $\mathbf{a.}$ The quantities $\cos \alpha $%
, $\cos \beta $, and $\cos \gamma $ are called the \textbf{direction cosines 
}of $\mathbf{a.}$ If $\mathbf{a}=\left\langle a_{1},a_{2},a_{3}\right\rangle 
$, using the fact that $\cos \theta =\frac{\mathbf{a}\cdot \mathbf{b}}{%
\left| \mathbf{a}\right| \left| \mathbf{b}\right| }$ and the fact that $%
\mathbf{i}$\textbf{, }$\mathbf{j}$\textbf{,} and\textbf{\ }$\mathbf{k}$ are
unit vectors, it follows that: 
\[
\begin{array}{lll}
\cos \alpha =\frac{a_{1}}{\left| \mathbf{a}\right| } & \quad \cos \beta =%
\frac{a_{2}}{\left| \mathbf{a}\right| } & \quad \cos \gamma =\frac{a_{3}}{%
\left| \mathbf{a}\right| }
\end{array}
\]
and by squaring each of the three equations above, and adding, it follows
that 
\[
\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma =1 
\]
Also 
\[
\frac{\mathbf{a}}{\left| \mathbf{a}\right| }=\left\langle \cos \alpha ,\cos
\beta ,\cos \gamma \right\rangle 
\]
That is, $\left\langle \cos \alpha ,\cos \beta ,\cos \gamma \right\rangle $
is the unit vector in the same direction as $\mathbf{a.}$

\subsubsection{Projections\label{Projections}}

Suppose $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ are representations
of two vectors $\mathbf{a}$ and $\mathbf{b}$ with the same initial point $P$%
. If $S$ is the foot of the perpendicular from $R$ to the line containing $%
\overrightarrow{PQ}$ then the vector with representation $\overrightarrow{PS}
$ is called the \textbf{vector projection} of $\mathbf{b}$ onto $\mathbf{a}$
and is denoted $\limfunc{proj}_{\mathbf{a}}\mathbf{b}$. The \textbf{scalar
projection} of $\mathbf{b}$ onto $\mathbf{a}$ is 
\[
\left| \limfunc{proj}\nolimits_{\mathbf{a}}\mathbf{b}\right| 
\]
The scalar projection of $\mathbf{b}$ onto $\mathbf{a}$ is also called the 
\textbf{component of} $\mathbf{b}$ \textbf{along} $\mathbf{a}$ and is
denoted $\limfunc{comp}_{\mathbf{a}}\mathbf{b}$\textbf{.} Note that 
\[
\limfunc{comp}\nolimits_{\mathbf{a}}\mathbf{b}=\left| \limfunc{proj}%
\nolimits_{\mathbf{a}}\mathbf{b}\right| =\left| \mathbf{b}\right| \cos
\theta 
\]
where $\theta $ is the angle between $\mathbf{a}$ and $\mathbf{b}$\textbf{.}
It follows that 
\begin{eqnarray*}
\limfunc{comp}\nolimits_{\mathbf{a}}\mathbf{b} &=&\frac{\mathbf{a}\cdot 
\mathbf{b}}{\left| \mathbf{a}\right| } \\
\limfunc{proj}\nolimits_{\mathbf{a}}\mathbf{b} &=&\left( \frac{\mathbf{a}%
\cdot \mathbf{b}}{\left| \mathbf{a}\right| ^{2}}\right) \mathbf{a}
\end{eqnarray*}

\subsection{Related Reference Topics}

{\small \hyperref{Projections onto lines}{}{}{MA03_02.tex#Projections Onto
Lines}}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Dot product}{}{}{../../Help/DM8-2.tex#Matrix product}}

{\small \hyperref{Norm}{}{}{../../Help/DM8-5.tex#Norm}}

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