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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Cross Product">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Fri Mar 28 15:33:55 1997}
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\section{Cross Product\label{Cross Product}}

For any two vectors $a=\left( a_{1},a_{2},a_{3}\right) $ and $b=\left(
b_{1},b_{2},b_{3}\right) $ in $\UNICODE{0x211d} ^{3}$, their \textbf{cross
product }$a\times b$ \label{cross product}is defined to be the vector 
\[
a\times b=\left(
a_{2}b_{3}-a_{3}b_{2},a_{3}b_{1}-a_{1}b_{3},a_{1}b_{2}-a_{2}b_{1}\right) 
\]

\begin{itemize}
\item  For a $3\times 3$ matrix, 
\[
\det \left[ 
\begin{array}{lll}
a_{1} & a_{2} & a_{3} \\ 
b_{1} & b_{2} & b_{3} \\ 
c_{1} & c_{2} & c_{3}
\end{array}
\right] =\left( a\times b\right) ^{T}c
\]
where $a=\left( a_{1},a_{2},a_{3}\right) $, $b=\left(
b_{1},b_{2},b_{3}\right) $, and $c=\left( c_{1},c_{2},c_{3}\right) $. 

\item  If $a$ and $b$ are two vectors in $\UNICODE{0x211d} ^{3}$ and $\theta 
$ is the angle between $a$ and $b$, then $\left| \left| a\times b\right|
\right| =\left| \left| a\right| \right| \left| \left| b\right| \right| \sin
\theta $ is equal to the area of the parallelogram determined by $a$ and $b$.

\item  Let $A=\left[ 
\begin{array}{lll}
a_{1} & a_{2} & a_{3} \\ 
b_{1} & b_{2} & b_{3} \\ 
c_{1} & c_{2} & c_{3}
\end{array}
\right] $. The absolute value of $\det A$ is equal to the volume of the
parallelopiped determined by the rows of $A$, $a=\left(
a_{1},a_{2},a_{3}\right) $, $b=\left( b_{1},b_{2},b_{3}\right) $, and $%
c=\left( c_{1},c_{2},c_{3}\right) $. That is, 
\[
\left| \det A\right| =\left| \left( a\times b\right) ^{T}c\right| 
\]

\item  The vector $a\times b$ is orthogonal (perpendicular) to both $a$ and $%
b$.$\bigskip $ That is, 
\[
\left( a\times b\right) \cdot a=\left( a\times b\right) \cdot b=0
\]
\end{itemize}

\subsection{\ Related Reference Topics}

{\small \hyperref{Cross product}{}{}{Cal10_04.tex}}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Area of parallelogram}{}{}{%
../../Help/DM9-1.tex#Parallelogram}}

{\small \hyperref{Computing inner product with matrix multiplication}{}{}{%
../../Help/DM8-2.tex#Matrix product}}

{\small \hyperref{Computing vector inner product}{}{}{%
../../Help/DM9-1.tex#Dot product}}

{\small \hyperref{Cross product}{}{}{../../Help/DM9-1.tex#Cross product}}

{\small \hyperref{Volume of parallelpiped}{}{}{%
../../Help/DM9-1.tex#Parallelpiped}}

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