ScalarFactorTransferRules 2005-08-04 15:17:35 2005-08-04 16:39:22 Topic Euclidean vector scalar multiplication % this is the default PlanetMath 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 \theoremstyle{definition} \newtheorem*{thmplain}{Theorem} The different kinds of products between two Euclidean vectors may \PMlinkescapetext{contain} an additional scalar as factor in either vector factor $\vec{u}$, $\vec{v}$.\, Then such a scalar $r$ can be transferred from a vector to the other vector and to the whole product.\, This is true for scalar product, $$\vec{u}\!\cdot\!(r\vec{v}) = (r\vec{u})\!\cdot\!\vec{v} = r(\vec{u}\!\cdot\!\vec{v}),$$ for vector product, $$\vec{u}\!\times\!(r\vec{v}) = (r\vec{u})\!\times\!\vec{v} = r(\vec{u}\!\times\!\vec{v}),$$ and also for dyad product, $$\vec{u}(r\vec{v}) = (r\vec{u})\vec{v} = r(\vec{u}\vec{v}).$$