vector subtraction and position vectorsVectorSubtractionAndPositionVectors2026-08-20 03:24:142026-08-20 03:26:25Definition% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands here\section*{Vector Subtraction and Position Vectors}
Vector subtraction is defined by addition of the negative:
\[
\mathbf{u}-\mathbf{v}=\mathbf{u}+(-\mathbf{v}).
\]
Equivalently, $\mathbf{u}-\mathbf{v}$ is the vector which, when added to
$\mathbf{v}$, gives $\mathbf{u}$.
\begin{center}
\includegraphics[width=.68\textwidth]{brand_fig_5a.png}
\textit{Figure 5a, modernized: vector addition and subtraction.}
\end{center}
If $O$ is a fixed origin, define the position vectors
\[
\mathbf{r}_A=\overrightarrow{OA},\qquad
\mathbf{r}_B=\overrightarrow{OB}.
\]
Then the displacement from $A$ to $B$ is
\[
\boxed{\overrightarrow{AB}=\mathbf{r}_B-\mathbf{r}_A.}
\tag{1}
\]
This relation is fundamental in mechanics: displacement is the difference of
position vectors.
\begin{center}
\includegraphics[width=.68\textwidth]{brand_fig_5b.png}
\textit{Figure 5b, modernized: displacement as the difference of position vectors.}
\end{center}
\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.