式展開: 四元数の逆元

\begin{array}{rcl} q \overset{―}{q} & = & (w, V) (w, - V) \\ \dots \overset{―}{q} = w - x i - y j - z k = (w, - V), (実 部, i j k 部), V = x i + y j + z k \\ = & (w w - V \cdot (- V), w (- V) + w V + V \times (- V)) \\ \dots q_{1} q_{2} = (w_{1}, V_{1}) (w_{2}, V_{2}) = (w_{1} w_{2} - V_{1} \cdot V_{2}, w_{1} V_{2} + w_{2} V_{1} + V_{1} \times V_{2}) \\ = & (w^{2} + | V |^{2}, 0) \dots A \cdot A = {| A |}^{2}, A \times A = 0 \end{array}

\begin{array}{r} | q |^{2} = q \overset{―}{q} = \overset{―}{q} q = w^{2} + | V |^{2} \end{array}

\begin{array}{rclrclr} \frac{q \overset{―}{q}}{| q |^{2}} & = & q \frac{\overset{―}{q}}{| q |^{2}} & = & \frac{\overset{―}{q} q}{| q |^{2}} & = & \frac{\overset{―}{q}}{| q |^{2}} q = 1 \end{array}

\begin{array}{rcl} q^{- 1} & = & \frac{\overset{―}{q}}{| q |^{2}} \dots A B = B A = E (単 位 元) と な る B を A^{- 1} (逆 元) と す る \end{array}

\begin{array}{rcl} q q^{- 1} & = & q \frac{\overset{―}{q}}{| q |^{2}} = \frac{1}{| q |^{2}} q \overset{―}{q} \\ = & \frac{1}{w^{2} + | V |^{2}} (w, V) (w, - V) \\ = & \frac{1}{w^{2} + | V |^{2}} (w^{2} + | V |^{2}, w (- V) + w V + V \times V) \\ = & \frac{1}{w^{2} + | V |^{2}} (w^{2} + | V |^{2}, 0) \\ = & (\frac{w^{2} + | V |^{2}}{w^{2} + | V |^{2}}, \frac{w^{2} + | V |^{2}}{w^{2} + | V |^{2}}) \\ = & (1, 0) (単 位 元) \end{array}

式展開