内積計算の基本則

• 交換法則
$\stackrel{\to }{a}·\stackrel{\to }{b}=\stackrel{\to }{b}·\stackrel{\to }{a}$

• 定数倍
$\left(k\stackrel{\to }{a}\right)·\stackrel{\to }{b}=\stackrel{\to }{a}·\left(k\stackrel{\to }{b}\right)=k\left(\stackrel{\to }{a}·\stackrel{\to }{b}\right)$

• 分配法則
$\left(\stackrel{\to }{a}+\stackrel{\to }{b}\right)·\stackrel{\to }{c}=\stackrel{\to }{a}·\stackrel{\to }{c}+\stackrel{\to }{b}·\stackrel{\to }{c}$

■導出計算

●交換法則

$\stackrel{\to }{a}·\stackrel{\to }{b}=|\stackrel{\to }{a}||\stackrel{\to }{b}|cos\theta =|\stackrel{\to }{b}||\stackrel{\to }{a}|cos\theta =\stackrel{\to }{b}·\stackrel{\to }{a}$

●定数倍

$\begin{array}{l}\begin{array}{ll}\left(k\stackrel{\to }{a}\right)·\stackrel{\to }{b}\hfill & =|k\stackrel{\to }{a}||\stackrel{\to }{b}|cos\theta \hfill \\ \hfill & =k|\stackrel{\to }{a}||\stackrel{\to }{b}|cos\theta \hfill \\ \hfill & =|\stackrel{\to }{a}||k\stackrel{\to }{b}|cos\theta \hfill \\ \hfill & =\stackrel{\to }{a}·\left(k\stackrel{\to }{b}\right)\hfill \end{array}\\ \begin{array}{ll}\left(k\stackrel{\to }{a}\right)·\stackrel{\to }{b}\hfill & =|k\stackrel{\to }{a}||\stackrel{\to }{b}|cos\theta \hfill \\ \hfill & =k|\stackrel{\to }{a}||\stackrel{\to }{b}|cos\theta \hfill \\ \hfill & =k\left(\stackrel{\to }{a}·\stackrel{\to }{b}\right)\hfill \end{array}\end{array}$

●分配法則

$\begin{array}{ll}\left(\stackrel{\to }{a}+\stackrel{\to }{b}\right)·\stackrel{\to }{c}\hfill & =|\stackrel{\to }{a}+\stackrel{\to }{b}||\stackrel{\to }{c}|cos\theta \hfill \\ \hfill & =\left(|\stackrel{\to }{a}+\stackrel{\to }{b}|cos\theta \right)|\stackrel{\to }{c}|\hfill \\ \hfill & =\left(|\stackrel{\to }{a}|cos\alpha +|\stackrel{\to }{b}|cos\beta \right)|\stackrel{\to }{c}|\hfill \\ \hfill & =|\stackrel{\to }{a}||\stackrel{\to }{c}|cos\alpha +|\stackrel{\to }{b}||\stackrel{\to }{c}|cos\beta \hfill \\ \hfill & =\stackrel{\to }{a}·\stackrel{\to }{c}+\stackrel{\to }{b}·\stackrel{\to }{c}\hfill \end{array}$
（右上図参照）

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