cross productCrossProcuct2006-07-22 11:35:542026-08-21 03:02:18DefinitionThe build was dying on:
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public-domain article
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% define commands hereThe cross product or vector product is defined by
$$ \mathbf{A} \times \mathbf{B} = \left ( A_y B_z - A_z B_y \right ) \mathbf{\hat{i}} + \left ( A_z B_x - A_x B_z \right ) \mathbf{\hat{j}} + \left ( A_x B_y - A_y B_x \right ) \mathbf{\hat{k}}$$
Like the dot product, it is useful to look at its geometric definition and properties. Instead of the cosine of the angle between the two vectors the cross product is defined geometrically as
$$ \mathbf{A} \times \mathbf{B} = \left | \mathbf{A} \right | \left | \mathbf{B} \right | \sin \theta \mathbf{\hat{n}} $$
It is important to see that the unit vector $\mathbf{\hat{n}}$ is normal to the plane defined by the two vectors with the direction determined by the right hand rule.
It can be easier to remember the definition of the cross product with the determinant formulation
$$ \mathbf{A} \times \mathbf{B} = \left| \begin{matrix}
\mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}} \\
A_x & A_y & A_z \\
B_x & B_y & B_z
\end{matrix}\right| = \left ( A_y B_z - A_z B_y \right ) \mathbf{\hat{i}} + \left ( A_z B_x - A_x B_z \right ) \mathbf{\hat{j}} + \left ( A_x B_y - A_y B_x \right ) \mathbf{\hat{k}}$$
\section*{Vector Product (Cross Product)}
For vectors $\mathbf{A},\mathbf{B}\in\mathbb R^3$ separated by the smaller angle
$\theta$, the cross product is defined by
\[
\boxed{\mathbf{A}\times\mathbf{B}
=\lVert\mathbf{A}\rVert\lVert\mathbf{B}\rVert\sin\theta\,\hat{\mathbf n},}
\tag{1}
\]
where $\hat{\mathbf n}$ is the unit normal selected by the right-hand rule.
\begin{center}
\includegraphics[width=.68\textwidth]{brand_fig_16.png}
\textit{Figure 16, modernized: right-hand-rule orientation of the cross product.}
\end{center}
Its magnitude equals the area of the parallelogram generated by the vectors:
\[
\lVert\mathbf{A}\times\mathbf{B}\rVert
=\lVert\mathbf{A}\rVert\lVert\mathbf{B}\rVert\sin\theta.
\]
For nonzero vectors,
\[
\mathbf{A}\times\mathbf{B}=\mathbf0
\quad\Longleftrightarrow\quad
\mathbf{A}\parallel\mathbf{B}.
\]
The cross product is anticommutative:
\[
\mathbf{B}\times\mathbf{A}=-\mathbf{A}\times\mathbf{B}.
\]
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 \PMlinkescape{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.