📚 Properties of Scalar Triple Products | 标量三重积的性质
In AQA A-Level Further Mathematics, the scalar triple product is a compact way of combining three vectors in 3D to produce a single number. It appears when calculating volumes, testing whether vectors are coplanar, and simplifying vector algebra.
在 AQA A-Level 进阶数学中，标量三重积是将三个三维向量组合成一个数值的简洁方法。它在计算体积、判断向量是否共面以及化简向量代数时经常出现。
1. Definition and Notation | 定义与记号
The scalar triple product of three vectors a, b and c is defined as a · (b × c). It is often written as [a, b, c].
三个向量 a、b 和 c 的标量三重积定义为 a · (b × c)，常记作 [a, b, c]。
[a, b, c] = a · (b × c)
The cross product b × c must be found first because the dot product requires two vectors. The expression a · b × c is understood to mean a · (b × c), not (a · b) × c.
必须先计算叉积 b × c，因为点积需要两个向量。表达式 a · b × c 通常理解为 a · (b × c)，而不是 (a · b) × c。
2. Geometric Meaning: Volume of a Parallelepiped | 几何意义：平行六面体的体积
The magnitude of the scalar triple product equals the volume of the parallelepiped spanned by the three vectors. A negative value simply means the three vectors form a left-handed system rather than a right-handed one.
标量三重积的绝对值等于三个向量张成的平行六面体的体积。负值只表示三个向量构成左手系，而不是右手系。
Volume = |a · (b × c)|
This geometric view explains why the scalar triple product is so useful in 3D geometry: it measures the amount of space enclosed by three vectors placed tail-to-tail.
这种几何观点解释了标量三重积在三维几何中为何如此有用：它度量了三个向量尾尾相接时所包围的空间大小。
3. Determinant Form | 行列式形式
If a = (a₁, a₂, a₃), b = (b₁, b₂, b₃) and c n as the determinant of a 3 × 3 matrix whose rows are the components of the three vectors.
如果 a = (a₁, a₂, a₃)，b = (b₁, b₂, b₃)，c = (c₁, c₂, c₃)，那么标量三重积可以写成以这三个向量的分量为行的 3 × 3 矩阵的行列式。
[a, b, c] = a₁(b₂c₃ − b₃c₂) + a₂(b₃c₁ − b₁c₃) + a₃(b₁c₂ − b₂c₁)
This determinant form is often the quickest way to compute the scalar triple product numerically in AQA exam questions.
在 AQA 考试题中，这种行列式形式通常是快速计算标量三重积数值的最快方法。
4. Cyclic Permutation Property | 循环排列性质
The scalar triple product is unchanged when the three vectors are cyclically permuted. That means rotating the order a, b, c does not change the value.
当三个向量循环排列时，标量三重积的值不变。也就是说，轮换顺序 a、b、c 不会改变它的值。
[a, b, c] = [b, c, a] = [c, a, b]
Equivalently, a · (b × c) = b · (c × a) = c · (a × b the fact that a cyclic permutation of rows in a 3 × 3 determinant is an even permutation.
等价地，a · (b × c) = b · (c × a) = c · (a × b)。这是因为 3 × 3 行列式中行的循环排列是偶排列。
5. Swapping Two Vectors Changes the Sign | 交换两个向量改变符号
If any two vectors are swapped, the scalar triple product changes sign. This is a direct consequence of the anticommutativity of the cross product.
如果交换任意两个向量，标量三重积会改变符号。这是叉积反交换律的直接结果。
[a, b, c] = −[a, c, b] = −[b, a, c] = −[c, b, a]
For example, a · (b × c) = −a · (c × b) because b × c = −c × b. This sign rule is essential when rearranging expressions in vector proofs.
例如，a · (b × c) = −a · (c × b)，因为 b × c = −c × b。在向量证明中重新排列表达式时，这个符号规则至关重要。
6. Interchanging Dot and Cross | 点积与叉积的互换
In a scalar triple product, the dot and cross can be interchanged without changing the value, provided the order of the vectors is preserved in the appropriate cyclic way.
在标量三重积中，点积和叉积可以互换而不改变值，只要向量的顺序以适当的循环方式保持。
(a × b) · c = a · (b × c)
This identity is true because the dot product is commutative, so (a × b) · c = c · (a × b), which is simply a cyclic permutation of the original scalar triple product.
这个恒等式成立是因为点积满足交换律，所以 (a × b) · c = c · (a × b)，这正好是原标量三重积的循环排列。
7. Coplanarity and Linear Dependence | 共面性与线性相关
Three vectors a, b and c are coplanar, or linearly dependent, exactly when their scalar triple product is zero.
三个向量 a、b 和 c 共面或线性相关，当且仅当它们的标量三重积为零。
a · (b × c) = 0 ⇔ a, b, c are coplanar or linearly dependent
Geometrically, if the three vectors lie in the same plane, the parallelepiped has zero height, so its volume is zero. This gives a simple algebraic test for coplanarity.
从几何上看，如果三个向量位于同一平面内，平行六面体的高度为零，因此体积为零。这为判断共面性提供了一个简单的代数方法。
8. Linearity in Each Argument | 对每个向量的线性性
The scalar triple product is linear in each of its three arguments. This means constants factor out and sums expand in any slot.
标量三重积对它的三个变量分别具有线性性。这意味着常数可以提出，和可以在任意位置展开。
[λa + μd, b, c] = λ[a, b, c] + μ[d, b, c]
Similar rules hold for the second and third vectors. This property follows from the linearity of both the dot product and the cross product.
第二个和第三个向量也满足类似的规则。这个性质来自点积和叉积的线性性。
For example, a · ((λb + μd) × c) = λ a · (b × c) + μ a · (d × c). This is useful when proving vector identities or solving equations involving unknown vectors.
例如，a · ((λb + μd) × c) = λ a · (b × c) + μ a · (d × c)。这在证明向量恒等式或求解含未知向量的方程时非常有用。
9. Zero Cases | 结果为零的情形
The scalar triple product is zero in several simple situations. If any one of the vectors is the zero vector, the product is immediately zero.
在几种简单情形下，标量三重积为零。如果其中任意一个向量是零向量，结果立即为零。
If two of the vectors are parallel, their cross product is the zero vector, so the scalar triple product is zero. Since b × c = 0 whenever b and c are parallel.
如果其中两个向量平行，它们的叉积为零向量，因此标量三重积为零。因为当 b 和 c 平行时，b × c = 0。
More generally, if a, b and c are coplanar, the volume of the parallelepiped is zero, so the scalar triple product is zero.
更一般地，如果 a、b 和 c 共面，平行六面体的体积为零，因此标量三重积为零。
10. Worked Example | 计算示例
Worked example: let a = (1, 2, 3), b = (4, 5, 6) and c = (7, 8, 10). First compute the cross product b × c.
计算示例：设 a = (1, 2, 3)，b = (4, 5, 6)，c = (7, 8, 10)。首先计算叉积 b × Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
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