dot productDotProduct2001-10-15 23:22:082006-03-10 17:09:13Definitionscalar square\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}Let $u=(u_1,u_2,\ldots,u_n)$ and $v=(v_1,v_2,\ldots,v_n)$ two vectors on $k^n$ where $k$ is a field (like $\mathbb{R}$ or $\mathbb{C}$).
Then we define the \emph{dot product} of the two vectors as:
$$u\cdot v=u_1v_1+u_2v_2+\cdots+u_nv_n.$$
Notice that $u\cdot v$ is NOT a vector but a scalar (an element from the field $k$).
If $u,v$ are vectors in $\mathbb{R}^n$ and $\vartheta$ is the angle between them, then we also have
$$u\cdot v=\Vert u\Vert\Vert v\Vert \cos\vartheta.$$
Thus, in this case, $u\perp v$ if and only if $u\cdot v=0$.
The special case\, $u \cdot u$\, of scalar product is the {\em scalar square} of the vector $u$.\, In $\mathbb{R}^n$ it equals to the square of the length of $u$:
$$u \cdot u = \Vert u \Vert^2$$