vector additionVectorAddition2026-08-20 03:11:372026-08-20 03:11:37Definition% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands hereIf successive displacements carry a point from $A$ to $B$ and then from $B$
to $C$, the net displacement is
\[
\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}.
\tag{1}
\]
This is the triangle rule for vector addition.
\begin{center}
\includegraphics[width=.68\textwidth]{brand_fig_3a.png}
\textit{Figure 3a, modernized: triangle/parallelogram rule for vector addition.}
\end{center}
Drawing both vectors from a common tail gives the equivalent parallelogram
rule. Vector addition satisfies
\[
\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}
\tag{2}
\]
and
\[
(\mathbf{u}+\mathbf{v})+\mathbf{w}
=\mathbf{u}+(\mathbf{v}+\mathbf{w}).
\tag{3}
\]
\begin{center}
\includegraphics[width=.68\textwidth]{brand_fig_3b.png}
\textit{Figure 3b, modernized: commutativity of vector addition.}
\end{center}
\begin{center}
\includegraphics[width=.68\textwidth]{brand_fig_3c.png}
\textit{Figure 3c, modernized: associativity of vector addition.}
\end{center}
The magnitude satisfies the triangle inequality
\[
\lVert\mathbf{u}+\mathbf{v}\rVert\le
\lVert\mathbf{u}\rVert+\lVert\mathbf{v}\rVert,
\]
with equality when the nonzero vectors point in the same direction.
For many vectors, place them head-to-tail in any order. The resultant joins
the initial point to the final point. A closed vector polygon has zero sum:
\[
\overrightarrow{AB}+\overrightarrow{BC}+\cdots+
\overrightarrow{GA}=\mathbf 0.
\]
\begin{center}
\includegraphics[width=.68\textwidth]{brand_fig_3d.png}
\textit{Figure 3d, modernized: vector polygon and closure.}
\end{center}
\section*{Problems}
\begin{enumerate}
\item For a regular hexagon $ABCDEF$ with center $O$, construct
$\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}+
\overrightarrow{AE}+\overrightarrow{AF}$ and compare it with
$\overrightarrow{AO}$.
\item For triangle $ABC$, let $P,Q,R$ be the midpoints of its sides and
choose any point $O$ in its plane. Construct and compare
\[
\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC},
\qquad
\overrightarrow{OP}+\overrightarrow{OQ}+\overrightarrow{OR}.
\]
\item For any quadrilateral $ABCD$, let $P,Q$ be the midpoints of diagonals
$AC,BD$, and $M$ the midpoint of $PQ$. Construct
\[
\overrightarrow{MA}+\overrightarrow{MB}+\overrightarrow{MC}+
\overrightarrow{MD}.
\]
\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.