dot product algebra and geometric applications DotProductAlgebraAndGeometricApplications 2026-08-21 02:53:08 2026-08-21 02:53:08 Example dot product % 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*{Dot-Product Algebra and Geometric Applications} The dot product is bilinear: \[ \mathbf{u}\cdot(\mathbf{v}+\mathbf{w}) =\mathbf{u}\cdot\mathbf{v}+\mathbf{u}\cdot\mathbf{w}, \] \[ (a\mathbf{u})\cdot(b\mathbf{v})=ab(\mathbf{u}\cdot\mathbf{v}). \] In Cartesian components, \[ \boxed{\mathbf{u}\cdot\mathbf{v}=u_xv_x+u_yv_y+u_zv_z.} \tag{1} \] \subsection*{Example 1} For $\mathbf{u}=(2,-1,3)$ and $\mathbf{v}=(0,2,4)$, \[ \mathbf{u}\cdot\mathbf{v}=10. \] Hence \[ \cos\theta=\frac{10}{\sqrt{14}\sqrt{20}} \] and $\theta\approx53.3^\circ$. \subsection*{Example 2: angle in a semicircle} If $AB$ is a diameter of a circle and $P$ lies on the circle, a vector proof gives \[ \overrightarrow{AP}\cdot\overrightarrow{BP}=0, \] so the inscribed angle $APB$ is a right angle. \begin{center} \includegraphics[width=.68\textwidth]{brand_fig_15a.png} \textit{Figure 15a, modernized: dot-product proof of Thales' theorem.} \end{center} \subsection*{Example 3: law of cosines} For triangle $ABC$, \[ \mathbf{a}=\mathbf{b}-\mathbf{c} \] implies \[ a^2=b^2+c^2-2bc\cos A. \] \begin{center} \includegraphics[width=.68\textwidth]{brand_fig_15b.png} \textit{Figure 15b, modernized: vector derivation of the law of cosines.} \end{center} \subsection*{Example 4: distance from a point to a line} For $A=(3,1,-1)$ and the line through $B=(2,3,0)$, $C=(-1,2,4)$, \[ \overrightarrow{AB}=(-1,2,1),\qquad \overrightarrow{BC}=(-3,-1,4). \] The scalar projection of $\overrightarrow{AB}$ along the line is \[ \frac{\overrightarrow{AB}\cdot\overrightarrow{BC}} {\lVert\overrightarrow{BC}\rVert}=\frac5{\sqrt{26}}. \] Therefore the perpendicular distance is \[ d=\sqrt{\lVert\overrightarrow{AB}\rVert^2- \left(\frac5{\sqrt{26}}\right)^2} =\sqrt{\frac{131}{26}}\approx2.245. \] \subsection*{Example 5: equation of a plane} A plane through $P_1=(x_1,y_1,z_1)$ with normal $\mathbf n=(A,B,C)$ satisfies \[ \boxed{\mathbf n\cdot(\mathbf r-\mathbf r_1)=0,} \] or \[ A(x-x_1)+B(y-y_1)+C(z-z_1)=0. \] Thus for a plane \[ Ax+By+Cz+D=0, \] $\mathbf n=(A,B,C)$ is a normal vector. \subsection*{Example 6: point-to-plane distance} The distance from $P_1=(x_1,y_1,z_1)$ to $Ax+By+Cz+D=0$ is \[ \boxed{d=\frac{|Ax_1+By_1+Cz_1+D|}{\sqrt{A^2+B^2+C^2}}.} \tag{2} \] \section*{Source problems} \begin{enumerate} \item Find the scalar component of $(2,3,-1)$ along the direction $(-1,-2,2)$. \item Find the component of $2\hat i+3\hat j+\hat k$ along a line in the first octant making equal angles with the coordinate axes. \item Find the component of $\hat i+2\hat j-\hat k$ along the direction $\hat i-\hat j+\hat k$. \item Find the angles between: (a) $(1,1,0)$ and $(1,0,1)$; (b) $(1,1,1)$ and $(1,0,0)$. \item Find the shortest distance from $A=(2,-3,-4)$ to the line through $B=(1,2,-3)$ and $C=(3,3,-5)$. \item Find the shortest distance from $A=(1,-2,1)$ to $4x-3y+12z-8=0$. \item Find the plane perpendicular to the line through $A=(3,4,-1)$ and $B=(5,2,3)$ at its midpoint. \item Find the angle between $x-y+z+2=0$ and $2x+y-z+1=0$. \item Prove the parallelogram law \[ \lVert\mathbf{a}+\mathbf{b}\rVert^2+ \lVert\mathbf{a}-\mathbf{b}\rVert^2 =2\lVert\mathbf{a}\rVert^2+2\lVert\mathbf{b}\rVert^2. \] \item Show that the sphere with diameter endpoints having position vectors $\mathbf{a},\mathbf{b}$ is described by \[ (\mathbf r-\mathbf{a})\cdot(\mathbf r-\mathbf{b})=0. \] \end{enumerate} \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.