VectorProductInGeneralVectorSpaces 2004-08-09 05:49:48 2004-08-09 05:57:00 Definition cross product % 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 The vector product can be defined in any finite dimensional vector space $V$ with $\dim V=n$. Let $v_1,\dots,v_n$ be a basis of $V$, we then define the vector product of the vectors $w_1,\dots,w_{n-1}$ in the following way: $$w_1\times\dots\times w_{n-1}=\sum_{j=1}^nv_j\det(w_1,\dots,w_{n-1},v_j).$$ One can easily see that some of the properties of the vector product are the same as in $\mathbb{R}^3$: \begin{itemize} \item If one of the $w_i$ is equal to $0$, then the vector product is $0$. \item If $w_i$ are linearly dependent, then the vector product is $0$. \item In a Euclidean vector space $w_1\times\dots\times w_{n-1}$ is perpendicular to all $w_i$. \end{itemize}