scalar triple productScalarTripleProduct2026-08-21 03:35:132026-08-21 03:35:13Definition% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands here\section*{Scalar Triple Product}
The scalar triple product is
\[
\boxed{[\mathbf{u},\mathbf{v},\mathbf{w}]
=(\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}.}
\tag{1}
\]
Its absolute value equals the volume of the parallelepiped generated by the
three vectors:
\[
V=|[\mathbf{u},\mathbf{v},\mathbf{w}]|.
\]
Its sign records orientation relative to a right-handed basis.
\begin{center}
\includegraphics[width=.68\textwidth]{brand_fig_18.png}
\textit{Figure 18, modernized: scalar triple product as oriented parallelepiped volume.}
\end{center}
The scalar triple product is invariant under cyclic permutation:
\[
(\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}
=(\mathbf{v}\times\mathbf{w})\cdot\mathbf{u}
=(\mathbf{w}\times\mathbf{u})\cdot\mathbf{v},
\]
and changes sign under exchange of any two vectors.
In Cartesian components,
\[
\boxed{
(\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}=
\begin{vmatrix}
u_x&u_y&u_z\\
v_x&v_y&v_z\\
w_x&w_y&w_z
\end{vmatrix}.}
\tag{2}
\]
Therefore nonzero vectors $\mathbf{u},\mathbf{v},\mathbf{w}$ are coplanar exactly
when their scalar triple product vanishes.
\subsection*{Example 1: volume}
For
\[
A=(-3,1,2),\ B=(-1,0,-2),\ C=(2,1,4),\ D=(2,-3,1),
\]
Brand obtains
\[
(\overrightarrow{AB}\times\overrightarrow{AC})\cdot
\overrightarrow{AD}=81.
\]
Thus the parallelepiped volume is $81$, and the tetrahedron volume is
$81/6=13.5$.
\subsection*{Example 2: line-plane intersection}
If a line is
\[
\mathbf r=\mathbf{r}_A+\lambda\mathbf d
\]
and a plane through $C,D,E$ has normal
\[
\mathbf n=(\mathbf r_D-\mathbf r_C)\times
(\mathbf r_E-\mathbf r_C),
\]
then the intersection parameter is
\[
\boxed{\lambda=
\frac{(\mathbf r_C-\mathbf{r}_A)\cdot\mathbf n}
{\mathbf d\cdot\mathbf n}.}
\]
For Brand's data
$A=(1,2,1)$, $B=(2,1,2)$, $C=(0,-4,4)$, $D=(2,-2,2)$,
$E=(4,1,2)$, this gives $\lambda=2$ and
\[
P=(3,0,3).
\]
\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 \PMlinkescape{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.