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Verification of the axioms for the ``geometric'' inner product in $\mathbb{R} ^2 $.

$\forall \overrightarrow{x} ,\overrightarrow{y}\in \mathbb{R} ^2$, $\langle \overrightarrow{x} ,\overrightarrow{y}\rangle
\overset{\text{def}}{=}\...
...tarrow{y}\end{Vmatrix}\cdot \cos \angle \overrightarrow{x} ,\overrightarrow{y} $.
1.
$\langle \overrightarrow{x} ,\overrightarrow{y}\rangle
= \begin{Vmatrix}\overri...
...w{y} ,\overrightarrow{x} =\langle \overrightarrow{y} ,\overrightarrow{x}\rangle$.
2.
(a)
$\underline{\langle \overrightarrow{x} ,\overrightarrow{x}\rangle}
=\begin{Vmat...
...s 0 = \begin{Vmatrix}\overrightarrow{x}\end{Vmatrix} ^2 \underline{\geqslant 0}$
(b)
$\langle \overrightarrow{x} ,\overrightarrow{x}\rangle=0
\Longleftrightarrow \be...
...x}\end{Vmatrix} = 0
\Longleftrightarrow \overrightarrow{x} =\overrightarrow{0} $.
3.
(a)

\begin{align*}\underline{\langle \overrightarrow{x} ,\overrightarrow{z}\rangle
...
...ightarrow{x} + \overrightarrow{y} ,\overrightarrow{z}\rangle }. \\
\end{align*}
(b)

\begin{align*}\underline{ \langle \alpha \overrightarrow{x} , \overrightarrow{y}...
... \alpha \langle \overrightarrow{x} ,\overrightarrow{y}\rangle }.\\
\end{align*}
In the previous calculations we used the following remark:

Remark 12.2.1 (1)   $\cos \angle \alpha \overrightarrow{x} ,\overrightarrow{y} =
\cos \angle \overrightarrow{x} ,\overrightarrow{y} $ if $\alpha > 0$; (2) $\cos \angle \alpha \overrightarrow{x} ,\overrightarrow{y} =
-\cos \angle \overrightarrow{x} ,\overrightarrow{y} $ if $\alpha < 0$. Thus, $\cos \angle \alpha \overrightarrow{x} ,\overrightarrow{y} =
\text{sgn}(\alpha)\cos \angle \overrightarrow{x} ,\overrightarrow{y} $.



Noah Dana-Picard
2001-02-26