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# 內積的幾何證明

<center><iframe scrolling="no" title="" src="https://www.geogebra.org/material/iframe/id/ebpc256r/width/800/height/500/border/888888/sfsb/false/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/true/ctl/false" width="600px" height="377px" style="border:0px;filter:invert(.9)"></iframe></center><br>

$$
令\cases{
\vec{u} = \vec{AB} = (u_x, u_y)\\
\vec{v} = \vec{AC} = (v_x, v_y)
}
$$

$$
\cases{
|u_x| = |AI| = |AJ|\\
|u_y| = |IB| = |JD|\\
|v_x| = |AF|\\
|v_y| = |FC|
}
$$

$$
\begin{array}l
\vec{u}\cdot\vec{v} &= |AB| \times (|AC|\times\cos(\angle CAH))\\
&= |AB| \times |AH| \\
&= |AD| \times |AH| \\
&=\text{ Area of } AGLD \\
&=\text{ Area of } AGKJ \\
&= AFMJ + FGKM \\
&= |AJ| \times |AF| + |AJ| \times |FG| \\
&= |u_x| \times |v_x| + |AJ| \times |FG|
\end{array}
$$

$$
\begin{array}l
\because
\triangle GFC \approx \triangle LKG\\
|LK|:|KG| = |GF|:|FC|\\
|KG|\times |GF| = |LK| \times |FC|\\
|AJ| \times |GF| = |JD| \times |FC|\\
|AJ| \times |GF| = |u_y| \times |v_y|
\end{array}
$$

$$
\begin{array}l
\therefore
\vec{u}\cdot\vec{v} &= |u_x| \times |v_x| + |AJ| \times |FG|\\
&= |u_x| \times |v_x| + |u_y| \times |v_y|
\end{array}
$$

###### tags: `geometry` `math`