📚 Triple Products of Vectors | 向量的三重积
In A-level Mathematics, triple products connect the two fundamental operations on vectors - the dot product and the cross product. They appear whenever three vectors are combined, and the result may be either a scalar or a new vector. A secure understanding of triple products makes problems about volumes, coplanarity and three-dimensional geometry far easier to handle.
在 A-level 数学中，三重积把向量的两种基本运算——点积与叉积——联系起来。只要把三个向量组合在一起，就会出现三重积，而结果可能是一个标量，也可能是一个新的向量。牢固掌握三重积，能让你更从容地处理体积、共面性和三维几何问题。
1. What Are Triple Products? | 什么是三重积？
A triple product is an expression built from three vectors. There are two main types. The scalar triple product combines a dot product with a cross product and produces a single number. The vector triple product uses two cross products and produces a new vector.
三重积是由三个向量构成的表达式，主要有两种类型。标量三重积把点积与叉积结合起来，得到一个数；向量三重积使用两次叉积，得到一个新的向量。
he cross product is not associative, so a × (b × c) is generally not equal to (a × b) × c.
标量三重积写作 a · (b × c)，向量三重积写作 a × (b × c)。括号非常重要：叉积不满足结合律，因此 a × (b × c) 一般不等于 (a × b) × c。
Typical uses include finding the volume of a parallelepiped, testing whether three vectors or four points are coplanar, and simplifying vector expressions in geometry and mechanics.
它们的典型用途包括：求平行六面体的体积、判断三个向量或四个点是否共面，以及化简几何与力学中的向量表达式。
2. Definition of the Scalar Triple Product | 标量三重积的定义
c₂, c₃), the scalar triple product is written with square brackets and defined as follows.
对于三个向量 a = (a₁, a₂, a₃)、b = (b₁, b₂, b₃)、c = (c₁, c₂, c₃)，标量三重积用方括号表示，定义如下。
[a, b, c] = a · (b × c)
n take the dot product with a, which gives a scalar. The notation [a, b, c] is standard in A-level vector work.
运算顺序是固定的：先算 b × c（这是一个向量），再与 a 作点积，得到一个标量。记号 [a, b, c] 是 A-level 向量部分的标准写法。
the cross may be interchanged as long as the cyclic order of the three vectors is preserved.
由于点积满足交换律，a · (b × c) = (b × c) · a。同时也有 a · (b × c) = (a × b) · c，因此只要保持三个向量的循环顺序不变，点积与叉积的位置就可以互换。
3. Determinant Form of the Scalar Triple Product | 标量三重积的行列式形式
e the components of a, b and c. Expanding the determinant gives a practical formula that is quick to evaluate by hand.
标量三重积可以写成一个 3 × 3 行列式，其各行分别是 a、b、c 的分量。展开这个行列式就得到一个便于手算的实用公式。
| a₁ | a₂ | a₃ |
| b₁ | b₂ | b₃ |
| c₁ | c₂ | c₃ |
Expanding the determinant along the first row gives the working formula used in most exam questions.
沿第一行展开这个行列式，就得到大多数考题中使用的计算公式。
[a, b, c] = a₁(b₂c₃ − b₃c₂) − a₂(b₁c₃ − b₃c₁) + a₃(b₁c₂ − b₂c₁)
A useful check: if any two rows of the determinant are identical or proportional, its value is zero. This gives an instant test for parallel vectors without any further calculation.
一个有用的检查方法：如果行列式的任意两行相同或成比例，其值就是零。这样无需进一步计算，就能立刻判断两个向量是否平行。
4. Geometric Meaning: Volume of a Parallelepiped | 几何意义：平行六面体的体积
Geometrically, |a · (b × c)| de equals the area of that base, so the volume is area multiplied by perpendicular height.
从几何上看，|a · (b × c)| 是以 a、b、c 为三条棱的平行六面体的体积。向量 b × c 垂直于底面的平行四边形，其模长等于底面积，因此体积等于底面积乘以垂直高度。
V = |a · (b × c)|
If the three vectors form a right-handed set in the order a, b, c, the triple product is positive; if the order is left-handed, the value is negative. This is exactly why the modulus is needed when the answer must be a volume.
如果三个向量按 a、b、c 的顺序构成右手系，混合积为正；若是左手系顺序，则为负。这正是求体积时必须取绝对值的原因。
The volume of the tetrahedron with vertices at the origin,
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