vector productVectorProduct22026-08-21 01:26:202026-08-21 01:26:20Definition% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands here\section*{Products of Two Vectors}
Two distinct vector products are fundamental in three-dimensional mechanics.
The \emph{dot product}
\[
\mathbf{u}\cdot\mathbf{v}
\]
returns a scalar and measures directional alignment. The \emph{cross product}
\[
\mathbf{u}\times\mathbf{v}
\]
returns a vector perpendicular to both inputs and encodes oriented area.
These products appear throughout mechanics:
\begin{itemize}
\item work: $W=\int\mathbf F\cdot d\mathbf r$,
\item power: $P=\mathbf F\cdot\mathbf{v}$,
\item torque: $\boldsymbol\tau=\mathbf r\times\mathbf F$,
\item angular momentum: $\mathbf L=\mathbf r\times\mathbf{p}$.
\end{itemize}
Vector division by a vector is not part of ordinary Euclidean vector algebra.
Division by a nonzero scalar remains ordinary scalar multiplication by its
reciprocal.
\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{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.