vectors in a planeVectorsInAPlane2026-08-20 04:38:122026-08-20 04:38:12Topic% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands here\section*{Vectors in a Plane}
Let $\mathbf{a}$ and $\mathbf{b}$ be nonparallel vectors in a plane. Every vector
$\mathbf{u}$ in that plane can be written uniquely as
\[
\boxed{\mathbf{u}=\alpha\mathbf{a}+\beta\mathbf{b}.}
\tag{1}
\]
Thus $\{\mathbf{a},\mathbf{b}\}$ is a basis for the plane.
\begin{center}
\includegraphics[width=.68\textwidth]{brand_fig_8.png}
\textit{Figure 8, modernized: decomposition of a planar vector in a two-vector basis.}
\end{center}
Uniqueness follows from linear independence: if
\[
\alpha\mathbf{a}+\beta\mathbf{b}=\mathbf0,
\]
then $\alpha=\beta=0$.
For noncollinear points $A,B,C$, every point $P$ in their plane has a unique
affine representation
\[
\boxed{\mathbf r_P=
\alpha\mathbf{r}_A+\beta\mathbf{r}_B+\gamma\mathbf r_C,
\qquad \alpha+\beta+\gamma=1.}
\tag{2}
\]
These are affine, or barycentric, coordinates relative to the triangle.
\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.