vectors in spaceVectorsInSpace2026-08-20 14:29:392026-08-21 14:05:24Topic% this is the default PhysicsLibrary preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here\section*{Vectors in Space}
Let $\mathbf{a},\mathbf{b},\mathbf{c}$ be three noncoplanar vectors. Every vector
$\mathbf{u}$ in three-dimensional Euclidean space has a unique representation
\[
\boxed{\mathbf{u}=\alpha\mathbf{a}+\beta\mathbf{b}+\gamma\mathbf{c}.}
\tag{1}
\]
Thus the three vectors form a basis of $\mathbb R^3$.
\begin{center}
\includegraphics[width=.68\textwidth]{brand_fig_9.png}
\textit{Figure 9, modernized: vector decomposition in a three-vector spatial basis.}
\end{center}
For four noncoplanar points $A,B,C,D$, every point $P$ can be represented
uniquely in affine form as
\[
\mathbf r_P=
\alpha\mathbf{r}_A+\beta\mathbf{r}_B+\gamma\mathbf r_C+
\delta\mathbf r_D,
\qquad
\alpha+\beta+\gamma+\delta=1.
\]
\section*{Source problem}
Through a point $P$ draw lines to the vertices $A,B,C,D$ of a tetrahedron,
meeting the opposite face planes at $K,L,M,N$. Using directed ratios, prove
that the sum of the four ratios in which $K,L,M,N$ divide $PA,PB,PC,PD$ is
$-1$.
\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 \PMlinkescapetext{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.