summary of vector algebraSummaryOfVectorAlgebra2026-08-22 18:01:132026-08-22 18:01:13TopicPMlinkescapetext had a capital letter that breaks pdf rendering% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands here\section*{Summary of Vector Algebra}
Summary of vector algebra techniques in introductory physics:
\subsection*{Basic operations}
\[
\overrightarrow{AB}=\mathbf{r}_B-\mathbf{r}_A,
\]
\[
\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u},
\qquad
(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w}),
\]
\[
\mathbf{u}-\mathbf{v}=\mathbf{u}+(-\mathbf{v}),
\]
\[
a(\mathbf{u}+\mathbf{v})=a\mathbf{u}+a\mathbf{v},
\qquad
(a+b)\mathbf{u}=a\mathbf{u}+b\mathbf{u}.
\]
\subsection*{Linear and affine representations}
If $\mathbf{a},\mathbf{b}$ are nonparallel, every vector in their plane is
$\alpha\mathbf{a}+\beta\mathbf{b}$. If
$\mathbf{a},\mathbf{b},\mathbf{c}$ are noncoplanar, every vector in space is
\[
\alpha\mathbf{a}+\beta\mathbf{b}+\gamma\mathbf{c}.
\]
For point division,
\[
\mathbf r_P=\frac{n\mathbf{r}_A+m\mathbf{r}_B}{m+n}.
\]
For weighted points,
\[
\mathbf r_G=\frac{\sum_i m_i\mathbf r_i}{\sum_i m_i}.
\]
\subsection*{Cartesian components}
\[
\mathbf{u}=u_x\hat i+u_y\hat j+u_z\hat k,
\qquad
\lVert\mathbf{u}\rVert^2=u_x^2+u_y^2+u_z^2.
\]
\subsection*{Dot product}
\[
\mathbf{u}\cdot\mathbf{v}
=\lVert\mathbf{u}\rVert\lVert\mathbf{v}\rVert\cos\theta
=u_xv_x+u_yv_y+u_zv_z.
\]
For nonzero vectors,
\[
\mathbf{u}\cdot\mathbf{v}=0\iff\mathbf{u}\perp\mathbf{v}.
\]
\subsection*{Cross product}
\[
\mathbf{u}\times\mathbf{v}
=\lVert\mathbf{u}\rVert\lVert\mathbf{v}\rVert\sin\theta\,\hat n,
\]
\[
\mathbf{u}\times\mathbf{v}=-\mathbf{v}\times\mathbf{u},
\]
\[
\mathbf{u}\times\mathbf{v}=
\begin{vmatrix}
\hat i&\hat j&\hat k\\
u_x&u_y&u_z\\
v_x&v_y&v_z
\end{vmatrix}.
\]
For nonzero vectors,
\[
\mathbf{u}\times\mathbf{v}=\mathbf0\iff\mathbf{u}\parallel\mathbf{v}.
\]
\subsection*{Triple products}
\[
(\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}
=\det[\mathbf{u}\ \mathbf{v}\ \mathbf{w}],
\]
whose absolute value is the parallelepiped volume, and
\[
\mathbf{u}\times(\mathbf{v}\times\mathbf{w})
=\mathbf{v}(\mathbf{u}\cdot\mathbf{w})-
\mathbf{w}(\mathbf{u}\cdot\mathbf{v}).
\]
These \PMlinkescapetext{identities} form the working vector toolkit used throughout the
remainder of \emph{Vectorial Mechanics}.
\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{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.