https://sichen-wang.github.io/tags/linear-algebra/Linear AlgebraHugoBlox Kit (https://hugoblox.com)en-usSun, 23 Feb 2025 20:24:25 +0000https://sichen-wang.github.io/media/icon_hu_9dfcb674aa885f38.pnghttps://sichen-wang.github.io/tags/linear-algebra/https://sichen-wang.github.io/blog/%E5%90%91%E9%87%8F%E4%B8%89%E9%87%8D%E7%A7%AF/Sun, 23 Feb 2025 20:24:25 +0000https://sichen-wang.github.io/blog/%E5%90%91%E9%87%8F%E4%B8%89%E9%87%8D%E7%A7%AF/<p>存在<strong>向量三重积</strong>公式 </p> \[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c} \]<hr> <p>设向量为： </p> \[ \mathbf{a} = (a_1, a_2, a_3), \quad \mathbf{b} = (b_1, b_2, b_3), \quad \mathbf{c} = (c_1, c_2, c_3). \]<p> </p> \[ \mathbf{b} \times \mathbf{c} = \begin{pmatrix} b_2 c_3 - b_3 c_2 \\ b_3 c_1 - b_1 c_3 \\ b_1 c_2 - b_2 c_1 \end{pmatrix} \]<p> </p> \[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \begin{pmatrix} a_2 (b_1 c_2 - b_2 c_1) - a_3 (b_3 c_1 - b_1 c_3) \\ a_3 (b_2 c_3 - b_3 c_2) - a_1 (b_1 c_2 - b_2 c_1) \\ a_1 (b_3 c_1 - b_1 c_3) - a_2 (b_2 c_3 - b_3 c_2) \end{pmatrix} \]<p>展开并整理各分量：</p> <ul> <li>第一分量：</li> </ul> \[ a_2 b_1 c_2 - a_2 b_2 c_1 - a_3 b_3 c_1 + a_3 b_1 c_3 = b_1 (a_2 c_2 + a_3 c_3) - c_1 (a_2 b_2 + a_3 b_3) \]<ul> <li>第二分量：</li> </ul> \[ a_3 b_2 c_3 - a_3 b_3 c_2 - a_1 b_1 c_2 + a_1 b_2 c_1 = b_2 (a_3 c_3 + a_1 c_1) - c_2 (a_3 b_3 + a_1 b_1) \]<ul> <li>第三分量：</li> </ul> \[ a_1 b_3 c_1 - a_1 b_1 c_3 - a_2 b_2 c_3 + a_2 b_3 c_2 = b_3 (a_1 c_1 + a_2 c_2) - c_3 (a_1 b_1 + a_2 b_2) \]<p>观察到每个分量中缺少 \(a_{1} b_{1} c_{1}\)（第一分量）、\(a_{2} b_{2} c_{2}\)（第二分量）、\(a_{3} b_{3} c_{3}\)（第三分量）</p> <p>添加这些项后，表达式可统一写为： </p> \[ \mathbf{b} (\mathbf{a} \cdot \mathbf{c}) - \mathbf{c} (\mathbf{a} \cdot \mathbf{b}) \]