equality of vectors EqualityOfVectors 2026-08-20 02:50:47 2026-08-20 02:50:47 Definition % 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 Two free vectors are equal when they have the same magnitude and the same direction. Thus \[ \mathbf{a}=\mathbf{b} \] means that one can translate either directed segment parallel to itself until it coincides with the other. For a parallelogram $ABCD$, \[ \overrightarrow{AB}=\overrightarrow{DC}, \qquad \overrightarrow{AD}=\overrightarrow{BC}. \] This is the geometric content behind treating a free vector independently of where it is drawn. For localized vectors, equality as free vectors does not by itself imply the same mechanical effect. More advanced mechanics distinguishes the vector value from its point of application or line of action. \section*{Source} This article is a modernized restatement of the Public Domain corresponding 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.