Properties of Scalar Triple Products | 标量三重积的性质

📚 Properties of Scalar Triple Products | 标量三重积的性质

The scalar triple product is a fundamental operation in vector algebra. For three vectors a, b and c, the scalar triple product is defined as a·(b×c) and is often denoted by [a, b, c]. This article explores its key properties, which are essential for AQA A-Level Mathematics.

标量三重积是向量代数中的一个基本运算。对于三个向量 a、b 和 c，标量三重积定义为 a·(b×c)，通常记作 [a, b, c]。本文探讨其关键性质，这些性质对 AQA A-Level 数学至关重要。

1. Definition and Notation | 定义与符号

For vectors a = (a₁, a₂, a₃), b = (b₁, b₂, b₃) and c = (c₁, c₂, c₃), the scalar triple product is initially defined as the dot product of a with the cross product b×c:

[a, b, c] = a·(b×c)

This produces a scalar (a real number), not a vector. The notation [a, b, c] is also called the box product or mixed product.

对于向量 a = (a₁, a₂, a₃)、b = (b₁, b₂, b₃) 和 c = (c₁, c₂, c₃)，标量三重积最初定义为 a 与叉积 b×c 的点积：

[a, b, c] = a·(b×c)

结果是一个标量（实数），而非向量。记号 [a, b, c] 也称作盒积或混合积。

2. Geometric Interpretation | 几何意义

The absolute value of the scalar triple product equals the volume of the parallelepiped formed by the three vectors. If the three vectors are edges of a parallelepiped, then its volume is |a·(b×c)|.

标量三重积的绝对值等于由这三个向量构成的平行六面体的体积。若三个向量是平行六面体的棱，则其体积为 |a·(b×c)|。

Base area: |b×c| (area of parallelogram)
Height: |a·n̂| where n̂ is the unit normal to the base
Volume = base area × height = |a·(b×c)|

底面积：|b×c|（平行四边形面积）
高：|a·n̂|，其中 n̂ 是底面的单位法向量
体积 = 底面积 × 高 = |a·(b×c)|

The sign of the scalar triple product indicates the orientation of the ordered triple (a, b, c). A positive value corresponds to a right-handed system; negative to a left-handed system.

标量三重积的符号表示有序三元组 (a, b, c) 的定向。正值对应右手系，负值对应左手系。

3. Cyclic Permutation | 循环置换

The scalar triple product is invariant under a cyclic permutation of the three vectors:

[a, b, c] = [b, c, a] = [c, a, b]

This means a·(b×c) = b·(c×a) = c·(a×b). A cyclic shift preserves the handedness of the triple, so the value remains unchanged.

标量三重积在向量的循环置换下保持不变：

[a, b, c] = [b, c, a] = [c, a, b]

即 a·(b×c) = b·(c×a) = c·(a×b)。循环移动保持三元组的定向不变，因此值不变。

4. Anti-cyclic Permutation and Sign Change | 反循环置换与符号变化

If the order is changed by a non-cyclic permutation, the sign flips. Thus:

[a, c, b] = −[a, b, c]

More generally, swapping any two adjacent vectors changes the sign. For example:

[a, c, b] = −[a, b, c]
[b, a, c] = −[a, b, c]
[c, b, a] = −[a, b, c]

如果顺序通过非循环置换改变，符号会翻转。因此：

[a, c, b] = −[a, b, c]

更一般地，交换任意两个相邻向量都会改变符号。例如：

[a, c, b] = −[a, b, c]
[b, a, c] = −[a, b, c]
[c, b, a] = −[a, b, c]

This property is equivalent to the antisymmetry of the determinant.

该性质等价于行列式的反对称性。

5. Effect of Swapping Two Vectors | 交换两个向量的效果

A single transposition (swap of any two vectors) always negates the scalar triple product. This is a direct consequence of the cross product antisymmetry and the dot product commutativity.

任意交换两个向量（一次对换）总是使标量三重积取相反数。这是叉积反对称性和点积交换律的直接结果。

For example, consider [a, b, c] = a·(b×c). Swapping b and c gives a·(c×b) = a·(−b×c) = −[a, b, c].

例如，考虑 [a, b, c] = a·(b×c)。交换 b 和 c 可得 a·(c×b) = a·(−b×c) = −[a, b, c]。

In determinant terms, this is exactly the effect of interchanging two rows or columns of the matrix whose rows are the vector components.

从行列式的角度看，这正好是交换以向量分量为行（或列）的矩阵中两行（或两列）的效果。

6. Scalar Multiplication | 标量乘法

Multiplying any one of the three vectors by a scalar λ multiplies the whole scalar triple product by λ:

[λa, b, c] = [a, λb, c] = [a, b, λc] = λ[a, b, c]

This follows from the scalar multiplication properties of the cross product and dot product. For example, a·((λb)×c) = a·(λ(b×c)) = λ a·(b×c).

将三个向量中的任意一个乘以标量 λ，整个标量三重积也乘以 λ：

[λa, b, c] = [a, λb, c] = [a, b, λc] = λ[a, b, c]

这可由叉积和点积的标量乘法性质推出。例如，a·((λb)×c) = a·(λ(b×c)) = λ a·(b×c)。

7. Linearity (Trilinearity) | 线性（多重线性）

The scalar triple product is linear in each vector separately. For example, linearity in the first argument means:

[a₁ + a₂, b, c] = [a₁, b, c] + [a₂, b, c]

Similarly, for the second and third arguments:

[a, b₁ + b₂, c] = [a, b₁, c] + [a, b₂, c]
[a, b, c₁ + c₂] = [a, b, c₁] + [a, b, c₂]

标量三重积对每个向量分别满足线性。例如，对第一个参数线性意味着：

[a₁ + a₂, b, c] = [a₁, b, c] + [a₂, b, c]

类似地，对第二个和第三个参数：

[a, b₁ + b₂, c] = [a, b₁, c] + [a, b₂, c]
[a, b, c₁ + c₂] = [a, b, c₁] + [a, b, c₂]

Combined with scalar multiplication, this makes the scalar triple product a multilinear (trilinear) functional.

结合标量乘法，标量三重积是一个多重线性（三线性）泛函。

8. Zero Product and Coplanarity | 零积与共面性

Three vectors are coplanar (linearly dependent) if and only if their scalar triple product is zero:

[a, b, c] = 0 ⇔ a, b, c are coplanar

This is because the parallelepiped has zero volume exactly when the three vectors lie in the same plane. In that case b×c is perpendicular to the plane, and a is also in the plane, so the dot product is zero.

三个向量共面（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 共面

这是因为当三个向量位于同一平面时，平行六面体的体积为零。此时 b×c 垂直于该平面，而 a 也在该平面内，因此点积为零。

For AOA A-Level, this is a powerful test for coplanarity of four points. Given points A, B, C, D, the vectors AB, AC and AD are coplanar if and only if [AB, AC, AD] = 0.

在 AQA A-Level 中，这是判断四个点是否共面的有力工具。给定点 A、B、C、D，向量 AB、AC 和 AD 共面当且仅当 [AB, AC, AD] = 0。

9. Determinant Form | 行列式形式

If the vectors are written as row vectors, the scalar triple product equals the determinant of the 3×3 matrix formed by them:

[a, b, c] = det ⎛⎝ a₁ a₂ a₃ ; b₁ b₂ b₃ ; c₁ c₂ c₃ ⎞⎠

In expanded form:

a·(b×c) = a₁(b₂c₃ − b₃c₂) − a₂(b₁c₃ − b₃c₁) + a₃(b₁c₂ − b₂c₁)

若将向量写成行向量，则标量三重积等于由它们构成的 3×3 矩阵的行列式：

[a, b, c] = det ⎛⎝ a₁ a₂ a₃ ; b₁ b₂ b₃ ; c₁ c₂ c₃ ⎞⎠

展开形式为：

a·(b×c) = a₁(b₂c₃ − b₃c₂) − a₂(b₁c₃ − b₃c₁) + a₃(b₁c₂ − b₂c₁)

This determinant interpretation immediately justifies all the permutation, linearity and sign properties discussed above.

这种行列式解释立即验证了前面讨论的所有置换、线性和符号性质。

10. Relation to Volume and Orientation | 体积与定向的关系

The signed volume of the parallelepiped is exactly the scalar triple product:

Signed volume = [a, b, c] = a·(b×c)

The absolute value |[a, b, c]| gives the ordinary volume. The sign is positive if the ordered triple forms a right-handed set, and negative if left-handed.

平行六面体的有向体积正好等于标量三重积：

有向体积 = [a, b, c] = a·(b×c)

绝对值 |[a, b, c]| 给出普通体积。如果有序三元组构成右手系，则符号为正；构成左手系则为负。

This is particularly useful when finding the volume of a tetrahedron with vertices O, A, B, C: the volume is (1/6) |[OA, OB, OC]|.

这在求以 O、A、B、C 为顶点的四面体体积时尤其有用：其体积为 (1/6) |[OA, OB, OC]|。

11. Application: Coplanarity and Vector Equations | 应用：共面性与向量方程

In vector geometry, the scalar triple product is frequently used to test whether three vectors are coplanar, to find whether four points lie in one plane, and to derive plane equations.

在向量几何中，标量三重积常用于判断三个向量是否共面、四个点是否在同一个平面内，以及推导平面方程。

Example: Determine whether points P(1, 2, 3), Q(2, 3, 4), R(3, 4, 5) and S(4, 5, 6) are coplanar.

示例：判断点 P(1, 2, 3)、Q(2, 3, 4)、R(3, 4, 5) 和 S(4, 5, 6) 是否共面。

Compute PQ = (1, 1, 1), PR = (2, 2, 2), PS = (3, 3, 3). Then

PQ·(PR×PS) = 1(2×3 − 2×3) − 1(2×3 − 2×3) + 1(2×3 − 2×3) = 0

计算 PQ = (1, 1, 1)、PR = (2, 2, 2)、PS = (3, 3, 3)。于是

PQ·(PR×PS) = 1(2×3 − 2×3) − 1(2×3 − 2×3) + 1(2×3 − 2×3) = 0

Since the scalar triple product is zero, the four points are coplanar (in fact collinear in this example).

由于标量三重积为零，这四个点共面（在本例中实际上共线）。

12. Summary of Key Properties | 性质总结

Here is a quick reference table of the most important properties for examinations.

以下是考试中最重要性质的快速参考表。

Property	Result
Cyclic permutation	[a, b, c] = [b, c, a] = [c, a, b]
Swap two vectors	[a, c, b] = −[a, b, c]
Scalar multiplication	[λa, b, c] = λ[a, b, c]
Coplanarity	[a, b, c] = 0 ⇔ coplanar
Volume of parallelepiped	V = \|[a, b, c]\|
Volume of tetrahedron	V = (1/6) \|[a, b, c]\|
Determinant form	[a, b, c] = det of the 3×3 matrix

掌握这些性质后，你就能在大学入学考试中熟练处理涉及标量三重积的向量问题。

Mastering these properties will allow you to handle vector problems involving scalar triple products with confidence in your A-level examinations.

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Properties of Scalar Triple Products | 标量三重积的性质