cross productCrossProcuct2006-07-22 11:35:542026-09-09 03:46:29Definitionremoved comment about tikz, file is in entry% Object 206: Cross Product
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{graphicx}The cross product (or vector product) of
$\mathbf{A}=(A_x,A_y,A_z)$ and $\mathbf{B}=(B_x,B_y,B_z)$ is
\[
\mathbf{A}\times\mathbf{B}
=
\left(A_yB_z-A_zB_y\right)\hat{\mathbf{i}}
+
\left(A_zB_x-A_xB_z\right)\hat{\mathbf{j}}
+
\left(A_xB_y-A_yB_x\right)\hat{\mathbf{k}}.
\]
Geometrically, its magnitude is
\[
\lVert \mathbf{A}\times\mathbf{B}\rVert
=
\lVert\mathbf{A}\rVert\,\lVert\mathbf{B}\rVert \sin\theta,
\]
where $\theta$ is the smaller angle between $\mathbf{A}$ and $\mathbf{B}$.
Its direction is perpendicular to the plane determined by the two vectors
and is selected by the right-hand rule:
\[
\mathbf{A}\times\mathbf{B}
=
\lVert\mathbf{A}\rVert\,\lVert\mathbf{B}\rVert \sin\theta\,
\hat{\mathbf{n}}.
\]
It is often convenient to remember the cross product through the
determinant mnemonic
\[
\mathbf{A}\times\mathbf{B}
=
\begin{vmatrix}
\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}}\\
A_x & A_y & A_z\\
B_x & B_y & B_z
\end{vmatrix}.
\]
\section*{Vector Product (Cross Product)}
For vectors $\mathbf{A},\mathbf{B}\in\mathbb{R}^3$ separated by the angle
$\theta$, one may summarize the geometric definition as
\begin{equation}
\mathbf{A}\times\mathbf{B}
=
\lVert\mathbf{A}\rVert\,\lVert\mathbf{B}\rVert\sin\theta\,\hat{\mathbf n}.
\end{equation}
\begin{center}
\includegraphics{Cross_Product_Right_Hand_Rule.png}
\textit{Figure: right-hand-rule orientation of the cross product.}
\end{center}
Its magnitude equals the area of the parallelogram spanned by
$\mathbf{A}$ and $\mathbf{B}$:
\[
\lVert\mathbf{A}\times\mathbf{B}\rVert
=
\lVert\mathbf{A}\rVert\,\lVert\mathbf{B}\rVert\sin\theta.
\]
For nonzero vectors,
\[
\mathbf{A}\times\mathbf{B}=\mathbf{0}
\quad\Longleftrightarrow\quad
\mathbf{A}\parallel\mathbf{B}.
\]
The cross product is anti-commutative:
\[
\mathbf{B}\times\mathbf{A}=-\,\mathbf{A}\times\mathbf{B}.
\]
It is a specifically three-dimensional Euclidean operation in this form.
\section*{Modern notation references}
The notation and terminology in this modernized article follow standard
present-day mechanics and vector-analysis usage, particularly:
\begin{enumerate}
\item J. R. Taylor, \emph{Classical Mechanics}, University Science Books, 2005.
\item D. Kleppner and R. Kolenkow, \emph{An Introduction to Mechanics},
2nd ed., Cambridge University Press, 2014.
\item H. Goldstein, C. Poole, and J. Safko, \emph{Classical Mechanics},
3rd ed., Addison--Wesley, 2002.
\end{enumerate}
\section*{Source}
This article is a modernized restatement of the corresponding
public-domain article in Louis Brand, \emph{Vectorial Mechanics},
John Wiley \& Sons, New York, 1930, Chapter I, ``Vector Algebra.''
The original 1930 edition is the source basis.
\section*{TikZ source for the figure}
A TikZ version of the figure has been provided separately in the file
\texttt{Cross\_Product\_Right\_Hand\_Rule\_tikz.tex}.