✖️外積

線性代數 ⟩ 向量 ⟩ 運算 ⟩ 外積 (🈯 同義詞："cross product")

  • 「平面外積」： u×v=∣u1u2v1v2∣\mathbf{u}\times\mathbf{v} = \begin{vmatrix} u_1 & u_2 \\ v_1 & v_2 \end{vmatrix}

  • 「空間外積」： u×v=(∣u2u3v2v3∣,∣u3u1v3v1∣,∣u1u2v1v2∣)=∣ijku1u2u3v1v2v3∣\mathbf{u}\times\mathbf{v} = \left( \begin{vmatrix} u_2 & u_3 \\ v_2 & v_3 \end{vmatrix} , \begin{vmatrix} u_3 & u_1 \\ v_3 & v_1 \end{vmatrix} , \begin{vmatrix} u_1 & u_2 \\ v_1 & v_2 \end{vmatrix} \right) = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}

  • 「四元數外積」： p×q=(s+u)×(t+v)=−sv+tu+(u×v)\mathbf{p} \times \mathbf{q} = ( {\color{orange}s} + \mathbf{u}) \times ( {\color{orange}t} + \mathbf{v}) = -{\color{orange}s} \mathbf{v} + {\color{orange}t} \mathbf{u} + (\mathbf{u} \times \mathbf{v})

  1. (反交換律)：u×v=−v×u\mathbf{u}\times\mathbf{v} = -\mathbf{v}\times\mathbf{u}

  2. (左分配律)：u×(v+w)=u×v+u×w\mathbf{u}\times (\mathbf{v} + \mathbf{w}) = \mathbf{u}\times\mathbf{v} + \mathbf{u}\times\mathbf{w} (右分配律也成立)

  3. (類結合律)：k(u×v)=(ku)×v=u×(kv)k(\mathbf{u}\times\mathbf{v}) = (k\mathbf{u})\times\mathbf{v} = \mathbf{u}\times(k\mathbf{v})

  4. v×v=0\mathbf{v}\times\mathbf{v}=0

  5. ∣u×v∣≤∥u∥∥v∥|\mathbf{u}\times\mathbf{v}| \le \|\mathbf{u}\| \|\mathbf{v}\|

  6. ∥u×v∥=∥u∥∥v∥sin⁡θ \|\mathbf{u}\times\mathbf{v}\|= \|\mathbf{u}\| \|\mathbf{v}\| \sin\theta

  7. u×v=0  ⟺  u∥v\mathbf{u}\times\mathbf{v} = \mathbf{0} \iff \mathbf{u} \parallel \mathbf{v} (👉 平行向量性質)

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