内積計算の基本則

交換法則
$3$\displaystyle{{a}\limits^{\rightarrow }\cdot {b}\limits^{\rightarrow }={b}\limits^{\rightarrow }\cdot {a}\limits^{\rightarrow }}$
定数倍
$3$\displaystyle{ $ k{a}\limits^{\rightarrow } $ \cdot {b}\limits^{\rightarrow }={a}\limits^{\rightarrow }\cdot $ k{b}\limits^{\rightarrow } $ =k $ {a}\limits^{\rightarrow }\cdot {b}\limits^{\rightarrow } $ }$
分配法則
$3$\displaystyle{ $ {a}\limits^{\rightarrow }+{b}\limits^{\rightarrow } $ \cdot {c}\limits^{\rightarrow }={a}\limits^{\rightarrow }\cdot {c}\limits^{\rightarrow }+{b}\limits^{\rightarrow }\cdot {c}\limits^{\rightarrow }}$
ベクトルの大きさと内積の関係
$3$\displaystyle{{\left|{{a}\limits^{\rightarrow }}\right|}^{2}={a}\limits^{\rightarrow }\cdot {a}\limits^{\rightarrow }}$

■導出計算

●交換法則

$3$\displaystyle{{a}\limits^{\rightarrow }\cdot {b}\limits^{\rightarrow }= \|{a}\limits^{\rightarrow }\| \|{b}\limits^{\rightarrow }\| \cos {\theta}= \|{b}\limits^{\rightarrow }\| \|{a}\limits^{\rightarrow }\| \cos {\theta}}$ $3$\displaystyle{={b}\limits^{\rightarrow }\cdot {a}\limits^{\rightarrow }}$

●定数倍

$3$\displaystyle{\begin{array}{lll}\begin{array}{lll} $ k{a}\limits^{\rightarrow } $ \cdot {b}\limits^{\rightarrow } & = \| k{a}\limits^{\rightarrow } \| \|{b}\limits^{\rightarrow }\| \cos {\theta} & \vspace{6}\\ & =k \|{a}\limits^{\rightarrow }\| \|{b}\limits^{\rightarrow }\| \cos {\theta} & \vspace{6}\\ & = \|{a}\limits^{\rightarrow }\| \| k{b}\limits^{\rightarrow } \| \cos {\theta} & \vspace{6}\\ & ={a}\limits^{\rightarrow }\cdot $ k{b}\limits^{\rightarrow } $ & \vspace{6}\\ \end{array}& \vspace{6}\\ \begin{array}{lll} $ k{a}\limits^{\rightarrow } $ \cdot {b}\limits^{\rightarrow } & = \| k{a}\limits^{\rightarrow } \| \|{b}\limits^{\rightarrow }\| \cos {\theta} & \vspace{6}\\ & =k \|{a}\limits^{\rightarrow }\| \|{b}\limits^{\rightarrow }\| \cos {\theta} & \vspace{6}\\ & =k $ {a}\limits^{\rightarrow }\cdot {b}\limits^{\rightarrow } $ & \vspace{6}\\ \end{array}& \vspace{6}\\ \end{array}}$

●分配法則

$分配$ $3$\displaystyle{ \left({ {a}\limits^{\rightarrow }+{b}\limits^{\rightarrow } }\right)\cdot {c}\limits^{\rightarrow } }$ $3$\displaystyle{ =\left|{ {a}\limits^{\rightarrow }+{b}\limits^{\rightarrow } }\right|\left|{{c}\limits^{\rightarrow }}\right|\cos {\theta} }$

$3$\displaystyle{ =\left({ \left|{ {a}\limits^{\rightarrow }+{b}\limits^{\rightarrow } }\right|\cos {\theta} }\right)\left|{{c}\limits^{\rightarrow }}\right| }$

$3$\displaystyle{ =\left({ \left|{{a}\limits^{\rightarrow }}\right|\cos \alpha +\left|{{b}\limits^{\rightarrow }}\right|\cos {\beta} }\right)\left|{{c}\limits^{\rightarrow }}\right| }$

$3$\displaystyle{ =\left|{{a}\limits^{\rightarrow }}\right|\left|{{c}\limits^{\rightarrow }}\right|\cos \alpha +\left|{{b}\limits^{\rightarrow }}\right|\left|{{c}\limits^{\rightarrow }}\right|\cos {\beta} }$

$3$\displaystyle{ ={a}\limits^{\rightarrow }\cdot {c}\limits^{\rightarrow }+{b}\limits^{\rightarrow }\cdot {c}\limits^{\rightarrow } }$

（図参照）

●ベクトルの大きさと内積の関係

内積の定義より

$3$\displaystyle{{a}\limits^{\rightarrow }\cdot {a}\limits^{\rightarrow }=\left|{{a}\limits^{\rightarrow }}\right|\left|{{a}\limits^{\rightarrow }}\right|\cos 0={\left|{{a}\limits^{\rightarrow }}\right|}^{2}}$

が得られる．

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最終更新日： 2025年10月11日