📚 Properties of the Vector Product | 向量积的性质
The vector product, also called the cross product, is an operation on two vectors in three-dimensional space. Unlike the scalar product, it returns a vector rather than a number, so its properties describe both a magnitude and a direction. This article summarises the definition, the algebraic rules and the geometric applications that A-level Mathematics students need for AQA, Edexcel and OCR vector questions, including areas of triangles and volumes of parallelepipeds.
向量积又称叉积，是对三维空间中的两个向量进行的一种运算。与标量积不同，它返回的是一个向量而不是一个数，因此其性质同时涉及大小和方向。本文系统总结定义、代数法则以及几何应用，涵盖 AQA、Edexcel 与 OCR 考纲中向量题目所需的全部要点，包括三角形面积与平行六面体体积的计算。
1. Definition and Notation | 向量积的定义与记号
The vector product of two vectors a and b is written as a × b and read as a cross b. It is defined only in three-dimensional space, and it produces another vector. In contrast, the scalar product a ⋅ b produces a scalar. Because the output is a vector, every calculation of a cross product must report both a magnitude and a direction, and marks are usually awarded separately for each of these.
两个向量 a 与 b 的向量积写作 a × b，读作 a 叉 b。它只在三维空间中有定义，并且产生另一个向量。与之相比，标量积 a ⋅ b 产生一个标量。由于结果是向量，任何叉积计算都必须同时给出大小和方向，考试评分时这两部分通常分别给分。
The defining relationship combines the magnitudes of the two vectors, the sine of the angle between them, and a unit vector perpendicular to both. Here θ is the angle between a and b, measured so that 0° ≤ θ ≤ 180°, and n̂ is the unit vector perpendicular to the plane containing a and b.
定义式把两个向量的模长、两者夹角的正弦以及一个同时垂直于两者的单位向量结合在一起。其中 θ 是 a 与 b 的夹角，满足 0° ≤ θ ≤ 180°；n̂ 是垂直于 a 与 b 所在平面的单位向量。
a × b = |a| |b| sin θ n̂
The order of the two vectors matters: swapping them reverses the direction of n̂, which is the first sign that the cross product is anti-commutative. Note also that the definition says nothing about the component form; the component formula is derived from this geometric statement using the right-handed unit vectors.
两个向量的书写顺序很重要：交换它们会使 n̂ 的方向反向，这正是叉积满足反交换律的第一个迹象。还要注意，定义式本身并不涉及坐标分量形式；分量公式是由这个几何定义借助右手系单位向量推导出来的。
2. Magnitude of the Vector Product | 向量积的模长
Taking the modulus of both sides of the definition gives the magnitude formula. Since n̂ is a unit vector, it has modulus 1 and disappears from the magnitude calculation. The magnitude therefore depends only on the lengths of the two vectors and the angle between them.
对定义式两边取模，就得到模长公式。由于 n̂ 是单位向量，其模长为 1，在计算模长时可以直接消去。因此模长只取决于两个向量的长度以及它们的夹角。
|a × b| = |a| |b| sin θ
tive, which is exactly what we expect from a length. This is an important contrast with the scalar product, where a ⋅ b = |a| |b| cos θ can be negative when the angle is obtuse.
由于在 0° ≤ θ ≤ 180° 范围内 sin θ 恒为非负，向量积的模长也恒为非负，这与长度的意义相符。这一点与标量积形成重要对比：标量积 a ⋅ b = |a| |b| cos θ 在夹角为钝角时可以为负。
. If the vectors are perpendicular, then θ = 90°, so sin θ = 1 and the magnitude reaches its maximum possible value |a| |b|.
由此可立即得到两个常考的特殊情形。若两个向量平行，则 θ = 0° 或 θ = 180°，sin θ = 0，模长为零。若两个向量垂直，则 θ = 90°，sin θ = 1，模长达到最大值 |a| |b|。
- Parallel: |a × b| = 0，the zero vector results. | 平行：结果为模长为 0 的向量。
- Perpendicular: |a × b| = |a| |b|，the maximum value. | 垂直：模长取到最大值。
- Same vector: a × a = 0，because θ = 0°. | 同一向量：因为夹角为 0°，故叉积为零向量。
3. Direction and the Right-Hand Rule | 方向与右手定则
The direction of a × b is perpendicular to both a and b, so the result is normal to the plane containing them. Two opposite directions are possible, and the right-hand rule selects the correct one. Point the fingers of your right hand along a and curl them towards b; your thumb then points along a × b.
a × b 的方向同时垂直于 a 和 b，因此结果垂直于两者所在的平面。垂直于平面有两个相反的方向，用右手定则确定正确的一个：右手指向 a 的方向，向 b 的方向弯曲四指，拇指所指即为 a × b 的方向。
Equivalently, the cross product obeys the right-handed screw rule: rotating a right-handed screw from a to b advances the screw along a × b. The standard basis vectors form a right-handed system, and this cyclic pattern is worth memorising because it generates every other product in the table of unit vectors.
等价地，叉积也服从右手螺旋定则：把右手螺旋从 a 转到 b，螺旋前进的方向就是 a × b 的方向。标准基向量构成右手系，这个循环规律值得牢记，因为它可以推出单位向量表中所有其他的乘积。
i × j = k, j × k = i, k × i = j
Reversing any of these three products changes the sign, so j × i = −k, and any vector crossed with itself gives the zero vector. In exam answers, always state the direction explicitly, for example as a multiple of a given unit normal vector.
把上述三个等式中的任意一个反序都会改变符号，例如 j × i = −k；任何向量与自身叉乘都得到零向量。考试作答时务必明确写出方向，例如写成某个给定单位法向量的倍数。
4. Component Form Using i, j and k | 单位向量的分量形式
For calculation, vectors are usually written in component form, and the cross product is expanded using distributivity together with the table of unit vector products. Writing a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, the result is the standard component formula.
实际计算时，向量通常写成分量形式，利用分配律和单位向量乘积表展开即可。设 a = a₁i + a₂j + a₃k，b = b₁i + b₂j + b₃k，则得到标准分量公式。
a × b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k
The middle component carries a minus sign, and forgetting it is the single most common error in this topic. Check your result by computing a ⋅ (a × b) and b ⋅ (a × b); both must equal zero, which confirms perpendicularity.
中间分量前带负号，忘记这个负号是本专题中最常见的错误。检验方法是计算 a ⋅ (a × b) 与 b ⋅ (a × b)，两者都必须等于 0，从而确认结果与原来的两个向量都垂直。
| × | i | j | k |
| i | 0 | k | −j |
| j | −k | 0 | i |
| k | j | −i | 0 |
5. The Determinant Method | 行列式计算方法
terminant whose first row contains the unit vectors and whose second and third rows contain the components of a and b. Expanding along the first row reproduces the component formula exactly and greatly reduces sign errors.
实际操作中，最稳妥的做法是写出一个 3 × 3 行列式：第一行是单位向量，第二、三行分别是 a 与 b 的分量。沿第一行展开即可得到分量公式，能大幅减少符号错误。
a × b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k
Worked example: take a = 2i + 3j − k and b = i − j + 4k ₁ = 2 × 4 − (−1)(1) = 9, and a₁b₂ − a₂b₁ = 2 × (−1) − 3 × 1 = −5, giving the result below.
例题：取 a = 2i + 3j − k，b = i − j + 4k。则 a₂b₃ − a₃b₂ = 3 × 4 − (−1)(−1) = 11，a₁b₃ − a₃b₁ = 2 × 4 − (−1)(1) = 9，a₁b₂ − a₂b₁ = 2 × (−1) − 3 × 1 = −5，于是得下式。
a × b = 11i − 9j − 5k
As a check, a ⋅ (a × b) = 2 × 11 + 3 × (−9) + (−1)(−5) = 22 − 27 + 5 = 0, and similarly for b, so the answer is consistent.
检验一下：a ⋅ (a × b) = 2 × 11 + 3 × (−9) + (−1)(−5) = 22 − 27 + 5 = 0，对 b 做同样计算也得到 0，说明答案正确。
6. Anti-Commutativity | 反交换律
Reversing the order of the vectors reverses the direction of the result but leaves the magnitude unchanged. This property is called anti-commutativity and is the clearest algebraic difference between the vector product and the scalar product.
交换两个向量的顺序会改变结果的方向，但不改变模长。这一性质称为反交换律，是向量积与标量积在代数上最明显的区别。
a × b = −(b × a)
The reason is geometric: the angle from b to a is the negative of the angle from a to b, and the right-hand rule therefore produces the opposite normal. Note that the scalar product is commutative, a ⋅ b = b ⋅ a, so the two products behave completely differently in this respect.
原因在于几何：由 b 转到 a 的角是由 a 转到 b 的角的相反数，右手定则因此给出相反的法向。而标量积满足交换律，a ⋅ b = b ⋅ a，可见两种乘积在这一性质上完全不同。
A useful consequence is that a × a = 0 for every vector, since swapping the two identical vectors must leave the value unchanged while also changing its sign, which forces the result to be the zero vector.
一个有用的推论是：对任意向量都有 a × a = 0。因为交换两个完全相同的向量，结果既应保持不变，又应变为相反数，这只可能是零向量。
7. Distributivity and Multiplication by a Scalar | 分配律与数乘
The vector product distributes over vector addition on both sides, so it behaves like ordinary multiplication when brackets are expanded. This is what allows the component formula to be derived from the geometric definition.
向量积对向量加法在左右两侧都满足分配律，因此展开括号时它的行为与普通乘法类似。正是这一性质使得分量公式可以从几何定义中推导出来。
a × (b + c) = a × b + a × c
(a + b) × c = a × c + b × c
A scalar multiple may be moved freely across the product, and it multiplies the magnitude but not the direction when positive. When the scalar is negative, the direction of the result also reverses.
数乘因子可以自由地在叉积中移动；当因子为正时，它只改变模长而不改变方向；当因子为负时，结果的方向也会反向。
(ka) × b = k(a × b) = a × (kb), k ∈ ℝ
These rules mean that the cross product is bilinear: linear in each argument separately. They are the reason a cross product can be expanded term by term exactly like a product of brackets.
这些法则说明叉积是双线性的：对每一个自变量都分别是线性的。这也是叉积可以像多项式乘法那样逐项展开的原因。
8. Non-Associativity | 不满足结合律
Unlike ordinary multiplication, the vector product is not associative, so the position of the brackets changes the answer. Writing (a × b) × c is therefore not the same as a × (b × c), and brackets must always be shown.
与普通乘法不同，向量积不满足结合律，括号的位置会改变结果。因此 (a × b) × c 与 a × (b × c) 不相等，书写时必须明确标出括号。
A quick demonstration uses the unit vectors. Since i × j = k and k × i = j, we obtain i × (i × j) = i × k = −j. On the other hand, (i × i) × j = 0 × j = 0, and the two results are clearly different.
用单位向量即可快速说明。由于 i × j = k，k × i = j，可得 i × (i × j) = i × k = −j。而 (i × i) × j = 0 × j = 0，两个结果显然不同。
Note also that a × (b × c) always lies in the plane of b and c, whereas (a × b) × c always lies in the plane of a and b. The vector triple product identity, a × (b × c) = (a ⋅ c)b − (a ⋅ b)c, is useful for checking such expressions.
还要注意，a × (b × c) 一定位于 b 与 c 所在的平面内，而 (a × b) × c 一定位于 a 与 b 所在的平面内。向量三重积恒等式 a × (b × c) = (a ⋅ c)b − (a ⋅ b)c 可用于检验这类表达式。
9. Parallel, Perpendicular and Collinear Tests | 平行、垂直与共线判定
ause then sin θ = 0. This is equivalent to saying that one vector is a scalar multiple of the other.
向量积给出了判定平行的决定性方法。两个非零向量平行，当且仅当它们的叉积为零向量，因为此时 sin θ = 0。这等价于说其中一个向量是另一个的标量倍数。
a × b = 0 ⇔ a is parallel to b (or one of them is 0)
By contrast, perpendicularity is tested with the scalar product: a ⋅ b up the two tests, so remember that sin goes with the cross product and cos with the dot product.
相对地，垂直性用标量积检验：当 cos θ = 0 时 a ⋅ b = 0。考生常常把两种检验混淆，记住正弦对应叉积、余弦对应点积即可。
- Parallel test: a × b = 0. | 平行判定：叉积为零向量。
- Perpendicular test: a ⋅ b = 0. | 垂直判定：点积为零。
- Collinear points: A, B and C are collinear if vector AB × vector AC = 0. | 三点共线：当向量 AB × 向量 AC = 0 时，A、B、C 共线。
10. Areas of Triangles and Parallelograms | 三角形与平行四边形面积
Geometrically, the magnitude of a × b equals the area of the parallelogram with adjacent sides a and b. This follows directly from the area formula base × perpendicular height, since |a| sin θ is exactly the perpendicular height to the base b.
从几何上看，|a × b| 等于以 a 与 b 为邻边的平行四边形的面积。这直接来自底 × 高的面积公式，因为 |a| sin θ 恰好是以 b 为底时的高。
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