scalar component and vector projection on an AxisScalarComponentAndVectorProjectionOnAnAxis2026-08-21 00:32:292026-08-21 00:32:29Definitionvector projection% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands here\section*{Scalar Component and Projection on an Axis}
Let $\hat{e}$ be a unit vector defining the positive direction of an
axis. The \emph{scalar component} of $\mathbf{u}$ along the axis is
\[
\operatorname{comp}_{\hat{e}}\mathbf{u}
=\lVert\mathbf{u}\rVert\cos\theta,
\tag{1}
\]
where $\theta$ is the angle from the positive axis to $\mathbf{u}$.
The corresponding \emph{vector projection} is
\[
\operatorname{proj}_{\hat{e}}\mathbf{u}
=(\lVert\mathbf{u}\rVert\cos\theta)\hat{e}.
\tag{2}
\]
After the dot product is introduced, these become
\[
\operatorname{comp}_{\hat{e}}\mathbf{u}
=\mathbf{u}\cdot\hat{e},
\qquad
\operatorname{proj}_{\hat{e}}\mathbf{u}
=(\mathbf{u}\cdot\hat{e})\hat{e}.
\]
\begin{center}
\includegraphics[width=.68\textwidth]{brand_fig_10.png}
\textit{Figure 10, modernized: signed scalar components of vectors on a directed axis.}
\end{center}
Projection is linear:
\[
\operatorname{comp}_{\hat{e}}(\mathbf{u}+\mathbf{v}+\mathbf{w})
=\operatorname{comp}_{\hat{e}}\mathbf{u}
+\operatorname{comp}_{\hat{e}}\mathbf{v}
+\operatorname{comp}_{\hat{e}}\mathbf{w}.
\]
\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.