scalar multiplication of vectorsScalarMultiplicationOfVectors2026-08-20 04:12:502026-08-20 04:12:50Definition% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands here\section*{Scalar Multiplication of Vectors}
For a real scalar $a$ and vector $\mathbf{u}$, the product $a\mathbf{u}$ has
magnitude
\[
\lVert a\mathbf{u}\rVert=|a|\,\lVert\mathbf{u}\rVert,
\]
and points in the same direction as $\mathbf{u}$ if $a>0$ and the opposite
direction if $a<0$.
The algebraic laws are
\[
(ab)\mathbf{u}=a(b\mathbf{u}),
\]
\[
(a+b)\mathbf{u}=a\mathbf{u}+b\mathbf{u},
\]
\[
a(\mathbf{u}+\mathbf{v})=a\mathbf{u}+a\mathbf{v}.
\]
For $a\ne0$,
\[
\frac{\mathbf{u}}{a}=\left(\frac1a\right)\mathbf{u}.
\]
Two nonzero vectors are parallel exactly when one is a scalar multiple of the
other:
\[
\mathbf{u}=k\mathbf{v}.
\]
In modern linear-algebra language, vector addition and scalar multiplication
make the set of free vectors into a vector space.
\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.