Remark 4

Let $F$ be a vector space of arbitrary dimension over the maximal ordered field $\mathbf{A}$, and let $\Psi$ be a positive non-degenerate symmetric bilinear form on $F$. Given two linearly independent vectors $x, y$ of $F$, let $F'$ be the vector plane which they generate; the angle of $x$ and $y$ is defined to be the angle of $x$ and $y$ considered as elements of the plane $F'$; it is denoted by $(\widehat{x, y})$. The cosine of this angle is, by virtue of (19), given by

(where $|x| = \Psi(x, x)^{1/2}$ is also called the length of the vector $x$), and is therefore independent of the orientation chosen on $F'$; the sine and the tangent of $(\widehat{x, y})$ change sign if one changes the orientation of $F'$. Given two non-zero proportional vectors $x, y$ of $F$, one sets $(\widehat{x, y}) = 0$ by convention.