混合积

混合积是指三个向量 ${a}$、${b}$ 和 ${c}$ 的数量积。它通常用来计算三个向量所定义的平行六面体的体积。混合积的定义如下：

$$({a} \times {b}) \cdot {c}$$

其中，${a} \times {b}$ 表示 ${a}$ 和 ${b}$ 的叉积（向量积），结果是一个向量；然后这个向量再与向量 ${c}$ 做点积（数量积），结果是一个标量。 可以用行列式来表达混合积：

$$({a} \times {b}) \cdot {c} = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{vmatrix}$$

其中，${a} = (a_1, a_2, a_3)$，${b} = (b_1, b_2, b_3)$，${c} = (c_1, c_2, c_3)$。

几何解释

混合积的绝对值等于由向量 ${a}$、${b}$ 和 ${c}$ 构成的平行六面体的体积

$$V = |({a} \times {b}) \cdot {c}|$$

性质

1. 交换性：
   • $({a} \times {b}) \cdot {c} = ({b} \times {c}) \cdot {a} = ({c} \times {a}) \cdot {b}$
2. 线性性：
   • 混合积对每个向量都是线性的，例如：

     $$(k {a} \times {b}) \cdot {c} = k ({a} \times {b}) \cdot {c}$$

     $$({a} \times ({b} + {c})) \cdot {d} = ({a} \times {b}) \cdot {d} + ({a} \times {c}) \cdot {d}$$

3. 平行向量：
   • 如果 ${a}$、${b}$ 和 ${c}$ 中有任意两个向量是平行的，那么混合积为零：

     $$({a} \times {b}) \cdot {a} = 0 \quad \text{如果} \quad {a} \parallel {b}$$

示例

data:
  # 基向量边
  - type: scatter3d
    mode: lines
    x: [0, 3]
    y: [0, 0]
    z: [0, 0]
    line: {width: 6, color: 'red'}   
  - type: scatter3d
    mode: lines
    x: [0, 1]
    y: [0, 2]
    z: [0, 0]
    line: {width: 6, color: 'red'}
  - type: scatter3d
    mode: lines
    x: [0, 1]
    y: [0, 0]
    z: [0, 2]
    line: {width: 6, color: 'red'}
  # 从a出发的边
  - type: scatter3d
    mode: lines
    x: [3, 4]
    y: [0, 2]
    z: [0, 0]
    line: {width: 6, color: 'black'}
  - type: scatter3d
    mode: lines
    x: [3, 4]
    y: [0, 0]
    z: [0, 2]
    line: {width: 6, color: 'black'}
  # 从b出发的边
  - type: scatter3d
    mode: lines
    x: [1, 4]
    y: [2, 2]
    z: [0, 0]
    line: {width: 6, color: 'black'}
  - type: scatter3d
    mode: lines
    x: [1, 2]
    y: [2, 2]
    z: [0, 2]
    line: {width: 6, color: 'black'}
  # 从c出发的边
  - type: scatter3d
    mode: lines
    x: [1, 4]
    y: [0, 0]
    z: [2, 2]
    line: {width: 6, color: 'black'}
  - type: scatter3d
    mode: lines
    x: [1, 2]
    y: [0, 2]
    z: [2, 2]
    line: {width: 6, color: 'black'}
  # 其他边
  - type: scatter3d
    mode: lines
    x: [2, 5]
    y: [2, 2]
    z: [2, 2]
    line: {width: 6, color: 'black'}
  - type: scatter3d
    mode: lines
    x: [4, 5]
    y: [2, 2]
    z: [0, 2]
    line: {width: 6, color: 'black'}
  - type: scatter3d
    mode: lines
    x: [4, 5]
    y: [0, 2]
    z: [2, 2]
    line: {width: 6, color: 'black'}
  - type: scatter3d
    mode: lines
    x: [5, 4]
    y: [2, 2]
    z: [2, 0]
    line: {width: 6, color: 'black'}
  - type: scatter3d
    mode: lines
    x: [5, 4]
    y: [2, 0]
    z: [2, 2]
    line: {width: 6, color: 'black'}
layout:
  scene:
    xaxis: {title: X, dtick: 1}
    yaxis: {title: Y, dtick: 1}
    zaxis: {title: Z, dtick: 1}
    aspectmode: cube