現在の数式のサイズは 3 です。 サイズを選択 2 3 4 5 6 7
 Tweet このページ

# 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 }}$

## ■導出計算

### ●交換法則

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

### ●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

$3\displaystyle{\begin{array}{lll} \( {a}\limits^{\rightarrow }+{b}\limits^{\rightarrow } \) \cdot {c}\limits^{\rightarrow } & = \| {a}\limits^{\rightarrow }+{b}\limits^{\rightarrow } \| \|{c}\limits^{\rightarrow }\| \cos {\theta} & \vspace{6}\\ & = \left({ \| {a}\limits^{\rightarrow }+{b}\limits^{\rightarrow } \| \cos {\theta} }\right) \|{c}\limits^{\rightarrow }\| & \vspace{6}\\ & = \left({ \|{a}\limits^{\rightarrow }\| \cos {\alpha}+ \|{b}\limits^{\rightarrow }\| \cos {\beta} }\right) \|{c}\limits^{\rightarrow }\| & \vspace{6}\\ & = \|{a}\limits^{\rightarrow }\| \|{c}\limits^{\rightarrow }\| \cos {\alpha}+ \|{b}\limits^{\rightarrow }\| \|{c}\limits^{\rightarrow }\| \cos {\beta} & \vspace{6}\\ & ={a}\limits^{\rightarrow }\cdot {c}\limits^{\rightarrow }+{b}\limits^{\rightarrow }\cdot {c}\limits^{\rightarrow } & \vspace{6}\\ \end{array}}$
（右上図参照）

ホーム>>カテゴリー分類>>ベクトル>>内積計算の基本