Definition – Dot product. – Math-reference.net

Definition

Let $\bf{x} = (x_1, ..., x_m)$ and $\bf{y} = (y_1, ..., y_m)$ be vectors in $\mathbb{R}^m$ . The dot product (or scalar product) of $\bf{y}$ and $\bf{x}$ is defined to be the real number

$\bf{x}\cdot\bf{y} = x_1 y_1 + ... + x_m y_m$

Properties of the dot product

Let $\textbf{x, y, z}\in{\mathbb{R}^m}$ , and $k, l\in{\mathbb{R}}$ . The dot product has the following properties,

[Symmetry]: $\bf{x}\cdot\bf{y} = \bf{y}\cdot\bf{x}$

[Linearity]: $(k\textbf{x} + l\textbf{y})\cdot\textbf{y} = k(\textbf{x}\cdot\textbf{z}) + k(\textbf{x}\cdot\textbf{z})$

[Positivity]: $\textbf{x}\cdot\textbf{x}>0,~\forall{\textbf{x}}\in{\mathbb{R}^m}\setminus{\{0\}}$