📚 Vector Products in Component Form | 向量积的分量形式
At A-level, the vector product is often introduced geometrically before being converted into an algebraic tool. In component form, it becomes a quick arithmetic recipe that lets you find a vector perpendicular to two given vectors without any diagram.
在 A-level 课程中，向量积通常先以几何方式引入，然后再转化为代数工具。采用分量形式后，向量积就成为一套快捷的算术规则，让你无需作图就能求出与两个已知向量都垂直的向量。
The vector product, also called the cross product, is written with a multiplication sign ×, for example a × b. It is defined only for three-dimensional vectors, which is exactly the context used in AQA A-level mathematics and further mathematics.
向量积又称叉积，用乘号 × 表示，例如 a × b。它只对三维向量定义，这正是 AQA A-level 数学与进阶数学中使用的场景。
1. The Geometric Definition | 向量积的几何定义
Given two vectors a and b that are not parallel, their vector product a × b is a vector perpendicular to both a and b. Its direction is given by the right-hand rule: if you curl the fingers of your right hand from a toward b, your thumb points in the direction of a × b.
给定两个不平行的向量 a 和 b，它们的向量积 a × b 是一个同时垂直于 a 和 b 的向量。其方向由右手定则确定：若用右手手指从 a 弯向 b，拇指所指的方向就是 a × b 的方向。
The magnitude of a × b is defined as |a||b|sinθ, where θ is the angle between a and b, with 0 ≤ θ ≤ 180°. This magnitude is equal to the area of the parallelogram formed by a and b.
向量积 a × b 的大小定义为 |a||b|sinθ，其中 θ 是 a 与 b 的夹角，0 ≤ θ ≤ 180°。这个大小等于由 a 与 b 所构成的平行四边形的面积。
If a and b are parallel, then sinθ = 0 and a × b = 0. If either vector is the zero vector, the vector product is also zero.
如果 a 与 b 平行，则 sinθ = 0，因此 a × b = 0。如果其中一个向量为零向量，向量积也等于零。
2. Definition in Component Form | 分量形式中的定义
Let a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k. The component form of the vector product is obtained by expanding the product using the distributive law and the cross-product rules for the unit vectors i, j and k.
设 a = a₁i + a₂j + a₃k，b = b₁i + b₂j + b₃k。向量积的分量形式可以通过使用分配律以及单位向量 i、j、k 的叉积规则展开求得。
The standard component formula for a × b is:
a × b 的标准分量公式为：
a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k
It is also common to write this as an ordered triple:
通常也写成有序三元组的形式：
a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)
This formula is the central result of this topic. You should memorise it or be able to reconstruct it quickly from the determinant form.
这个公式是本主题的核心结果。你应该记住它，或者能够通过行列式形式迅速推导出来。
3. The Determinant Form | 行列式记法
An efficient way to remember the component formula is to use a symbolic 3 × 3 determinant. The first row contains the unit vectors i, j and k; the second row contains the components of a; the third row contains the components of b.
记住分量公式的一种有效方法是使用一个符号化的 3 × 3 行列式。第一行是单位向量 i、j、k；第二行是 a 的分量；第三行是 b 的分量。
| i j k |
| a₁ a₂ a₃ |
| b₁ b₂ b₃ |
Expanding this determinant along the first row gives exactly the formula from the previous section. The i-component uses the submatrix formed from a₂, a₃, b₂ and b₃; the j-component has a minus sign because of the alternating sign pattern of determinants.
沿着第一行展开这个行列式，就正好得到上一节给出的公式。i 分量使用由 a₂、a₃、b₂、b₃ 组成的子矩阵；j 分量前面有负号，这是因为行列式展开具有正负交替的符号规律。
The determinant notation is usually acceptable in exams and is often quicker than writing out the full vector product formula. Be careful to keep the components of a and b in the correct rows.
在考试中通常可以使用行列式记法，而且它通常比写出完整向量积公式更快。注意保持 a 和 b 的分量位于正确的行中。
4. Properties of the Vector Product | 向量积的性质
The vector product obeys several important algebraic properties. First, it is anti-commutative:
向量积满足若干重要的代数性质。首先，它是反交换的：
a × b = −(b × a)
This means that exchanging the order of the two vectors reverses the direction of the result. Therefore, order matters in a vector product, unlike in a scalar product.
这意味着交换两个向量的顺序会使结果方向相反。因此，在向量积中顺序很重要，这与数量积不同。
Second, the vector product is distributive over vector addition:
其次，向量积对向量加法满足分配律：
a × (b + c) = a × b + a × c
Third, scalar multiples can be brought out of the product:
第三，数量倍数可以提到叉积外面：
(λa) × b = λ(a × b)
The vector product is not associative in general, so (a × b) × c is not usually equal to a × (b × c). This is a common false assumption to avoid.
一般来说，向量积不满足结合律，所以 (a × b) × c 通常不等于 a × (b × c)。这是一个需要避免的常见错误假设。
5. Vector Products of Unit Vectors | 单位向量的向量积
The vector products of the standard unit vectors form the basis of the component formula. Using the right-hand rule and the definition of magnitude, we obtain the following cyclic pattern:
标准单位向量的向量积构成分量公式的基础。利用右手定则和模长的定义，可以得到如下循环规律：
i × j = k, j × k = i, k × i = j
Reversing the order introduces a negative sign:
交换顺序会产生负号：
j × i = −k, k × j = −i, i × k = −j
Any unit vector multiplied by itself gives the zero vector:
任何单位向量与自身作叉积都得到零向量：
i × i = j × j = k × k = 0
These rules, together with the distributive law, produce the component formula. In particular, terms such as a₁b₂(i × j) and a₂b₁(j × i) combine to give the k-component.
这些规则连同分配律一起可以推出分量公式。特别地，像 a₁b₂(i × j) 和 a₂b₁(j × i) 这样的项会合并得到 k 分量。
6. The Area of a Parallelogram and a Triangle | 平行四边形与三角形的面积
One of the most important applications of the vector product is the calculation of areas. The magnitude of a × b equals the area of the parallelogram with adjacent sides a and b.
向量积最重要的应用之一是计算面积。a × b 的模长等于以 a 和 b 为邻边的平行四边形的面积。
Area of parallelogram = |a × b|
Since the area of a triangle is half the area of the parallelogram on the same base, the area of the triangle with sides a and b is:
因为三角形的面积等于同底等高平行四边形面积的一半，所以以 a 和 b 为边的三角形面积为：
Area of triangle = ½|a × b|
When coordinates are given, you should first form the two direction vectors from the same vertex, then compute their vector product and finally take half of its magnitude for a triangle.
当给定坐标时，你应该先从同一个顶点构造两个方向向量，然后计算它们的向量积，最后取模长的一半作为三角形的面积。
7. Finding a Normal Vector to a Plane | 求平面的法向量
Because a × b is perpendicular to both a and b, it is an ideal tool for finding a normal vector to a plane. If a plane contains two non-parallel direction vectors d₁ and d₂, then a normal vector n is simply:
因为 a × b 同时垂直于 a 和 b，所以它是求平面法向量的理想工具。如果平面包含两个不平行方向向量 d₁ 和 d₂，那么一个法向量 n 就是：
n = d₁ × d₂
For a plane with equation ax + by + cz = d, the vector (a, b, c) is a normal vector. Therefore, if you compute a vector product of two direction vectors in the plane, you can immediately write down the coefficients of the plane equation.
对于方程 ax + by + cz = d 的平面，向量 (a, b, c) 就是一个法向量。因此，如果你计算平面内两个方向向量的向量积，就可以直接写出平面方程的系数。
If a unit normal vector is required, divide n by its magnitude:
如果需要单位法向量，则将 n 除以其模长：
n̂ = (d₁ × d₂) / |d₁ × d₂|
This is especially useful in questions about angles between planes or distances from points to planes.
这种方法在求平面夹角或点到平面距离的题目中尤其有用。
8. Worked Example | 完整例题
Let a = 2i − 3j + k and b = i + 2j − 2k. Find a × b, the area of the parallelogram with sides a and b, and a unit vector perpendicular to both a and b.
设 a = 2i − 3j + k，b = i + 2j − 2k。求 a × b、以 a 和 b 为边的平行四边形的面积，以及同时垂直于 a 和 b 的单位向量。
Using the component formula:
使用分量公式：
a₂b₃ − a₃b₂ = (−3)(−2) − (1)(2) = 6 − 2 = 4
a₃b₁ − a₁b₃ = (1)(1) − (2)(−2) = 1 + 4 = 5
a₁b₂ − a₂b₁ = (2)(2) − (−3)(1) = 4 + 3 = 7
Therefore:
因此：
a × b = 4i + 5j + 7k
The area of the parallelogram is |a × b| = √(4² + 5² + 7²) = √90 = 3√10.
平行四边形的面积为 |a × b| = √(4² + 5² + 7²) = √90 = 3√10。
A unit perpendicular vector is (4i + 5j + 7k) / √90. You can check perpendicularity by computing dot products: a · (a × b) = 8 − 15 + 7 = 0, and b · (a × b) = 4 + 10 − 14 = 0.
单位垂直向量为 (4i + 5j + 7k) / √90。你可以通过点积检验垂直性：a · (a × b) = 8 − 15 + 7 = 0，b · (a × b) = 4 + 10 − 14 = 0。
9. Scalar Triple Product and Volume | 三重积与体积
The scalar triple product is defined as a · (b × c). Because b × c is a vector perpendicular to both b and c, the scalar product with a measures the component of a along that normal direction.
数量三重积定义为 a · (b × c)。因为 b × c 是同时垂直于 b 和 c 的向量，所以它与 a 的点积衡量了 a 沿该法线方向的分量。
The absolute value of the scalar triple product gives the volume of the parallelepiped formed by a, b and c:
数量三重积的绝对值给出由 a、b、c 构成的平行六面体的体积：
V = |a · (b × c)|
In component form, this quantity can be computed using a 3 × 3 determinant with rows a, b and c. If the determinant is zero, the three vectors are coplan
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