📚 The Vector Product | 向量积
The vector product, also called the cross product, is one of the two ways of multiplying two vectors. It applies to vectors in three-dimensional space and returns a new vector rather than a scalar. That new vector is perpendicular to both of the original vectors, and its length depends on the angle between them. In the AQA Further Mathematics specification the vector product is used to find the areas of triangles and parallelograms, to build the equation of a plane, to calculate angles between lines and planes, and to find perpendicular distances. This article covers the definition and the right-hand rule, the component formula, the products of the unit vectors, the algebraic properties, the main geometric applications and the standard exam techniques.
向量积（又称叉积）是两种向量乘法之一。它适用于三维空间中的向量，得到的结果是一个新向量而不是一个标量。这个新向量同时垂直于原来的两个向量，其长度取决于两者之间的夹角。在 AQA 进阶数学大纲中，向量积被用于求三角形与平行四边形的面积、建立平面的方程、计算直线与平面之间的夹角，以及求垂线距离。本文涵盖定义与右手定则、分量公式、单位向量的叉积、代数性质、主要几何应用以及标准考试技巧。
1. Definition and Right-Hand Rule | 定义与右手定则
mber, and that vector is perpendicular to both a and b. The magnitude of a × b is given by |a||b| sin θ, where θ is the angle between a and b, measured so that 0° ≤ θ ≤ 180°.
向量积写作 a × b，定义在三维空间中的两个向量 a 与 b 上。它返回的是一个向量而不是一个数，并且该向量同时垂直于 a 和 b。a × b 的大小由 |a||b| sin θ 给出，其中 θ 是 a 与 b 之间的夹角，且满足 0° ≤ θ ≤ 180°。
quivalently, the ordered triple (a, b, a × b) is right-handed, exactly like the coordinate axes (i, j, k).
方向由右手定则确定：如果右手四指从 a 向 b 弯曲，那么拇指所指的方向就是 a × b 的方向。等价地说，有序三元组 (a, b, a × b) 构成右手系，就像坐标轴 (i, j, k) 一样。
a × b = |a||b| sin θ n̂
Here n̂ el, then θ = 0° or θ = 180°, so sin θ = 0 and a × b = 0. Also, if either a or b is the zero vector, then a × b = 0.
这里的 n̂ 是同时垂直于 a 与 b 的单位向量，其方向由右手定则确定。由此立刻得到两个结论。如果 a 与 b 平行，则 θ = 0° 或 θ = 180°，于是 sin θ = 0，故 a × b = 0。另外，如果 a 或 b 是零向量，同样有 a × b = 0。
2. Component Form in Three Dimensions | 三维分量形式
₁i + b₂j + b₃k, the cross product is found component by component using the formula below.
实际计算中我们并不用夹角来求向量积，而是使用分量。如果 a = a₁i + a₂j + a₃k，b = b₁i + b₂j + b₃k，则叉积可以按下面的公式逐分量求出。
a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k
ins the unit vectors i, j and k, whose second row contains the components of a, and whose third row contains the components of b.
该公式来自展开一个 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 |
The safest way to remember the pattern is to notice the cyclic order: the indices run 2, 3 then 1, 2 then 1, 3, and the middle term (j) is the one with the reversed sign. Almost every lost mark in this topic comes from a sign slip in that j component.
记住这个规律最可靠的方法是利用循环顺序：下标依次为 2、3，然后 1、2，然后 1、3；而中间那一项（j 分量）是唯一取相反符号的一项。这个考点上丢分的原因几乎都是 j 分量的符号写错。
3. Products of the Unit Vectors | 单位向量的叉积
e cycle i → j → k → i gives a positive unit vector; moving backwards gives a negative one.
单位向量 i、j、k 构成右手系，它们的叉积按循环规律重复。沿循环 i → j → k → i 正向移动得到正的第三个单位向量；反向移动则得到负号。
i × j = k, j × k = i, k × i = j
g the cross product of a unit vector with itself always gives the zero vector, because the two vectors are parallel.
交换顺序会改变符号，因此 j × i = −k，k × j = −i，i × k = −j。单位向量与自身的叉积恒为零向量，因为这两个向量平行。
i × i = j × j = k × k = 0
These nine results, together with the distributive law, are enough to derive the full component formula in Section 2 without ever quoting a determinant.
这九个结果加上分配律，就足以在不引用行列式的情况下推导出第 2 节的完整分量公式。
4. Algebraic Properties | 代数性质
The vector product behaves differently from ordinary multiplication and from the scalar product, so it is worth listing its rules carefully before using it in algebra.
向量积的运算规律与普通乘法以及数量积都不同，因此在用于代数运算之前，有必要仔细列出它的法则。
Anti-commutative: × b = −(b × a). Order matters, and swapping the vectors flips the direction of the result.
反交换律：a × b = −(b × a)。顺序很重要，交换两个向量会反转结果的方向。
Distributive: a × (b + c) = a × b + a × c, and (a + b) × c = a × c + b × c.
分配律：a × (b + c) = a × b + a × c，且 (a + b) × c = a × c + b × c。
Scalar multiples: (λa) × b = λ(a × b) = a × (λb) for any real number λ.
与数乘相容：对任意实数 λ，(λa) × b = λ(a × b) = a × (λb)。
Zero result: nerally a × b = 0 if and only if a and b are parallel (or one of them is the zero vector).
结果为零：a × a = 0；更一般地，a × b = 0 当且仅当 a 与 b 平行（或其中一个是零向量）。
Not associative: in general (a × b) × c is not equal to a × (b × c), so brackets can never be dropped.
不满足结合律：一般地 (a × b) × c 不等于 a × (b × c)，因此括号绝不能省略。
Perpendicularity check: (a × b) = 0 and b · (a × b) = 0. This is a very quick way to verify an answer in an exam.
垂直性检验：a · (a × b) = 0 且 b · (a × b) = 0。这是考场上检查答案是否正确的极快方法。
5. Magnitude, Areas of Triangles and Parallelograms | 模长、三角形与平行四边形面积
ector product has a direct geometric meaning. For two vectors a and b placed tail to tail, |a × b| with sides a and b, and half of that value is the area of the triangle formed by a and b.
向量积的模长有直接的几何意义。对于起点重合的两个向量 a 和 b，|a × b| 等于以 a、b 为邻边的平行四边形的面积，而它的一半就是以 a、b 为两边的三角形的面积。
Area of parallelogram = |a × b| = |a||b| sin θ
Area of triangle = ½|a × b|
olumn vectors, then compute the cross product and halve its magnitude. Always use the vectors from the same vertex, otherwise the sign of the cross product changes but the magnitude, and therefore the area, does not.
对于顶点为 A、B、C 的三角形，取 a = AB、b = AC（都写成列向量），计算叉积后取模长的一半即可。注意两条边要从同一个顶点出发；否则叉积会改变符号，但模长即面积不会改变。
6. Finding a Unit Normal Vector | 求单位法向量
a and b, it is a normal vector to any plane that contains, or is parallel to, the directions a and b. Dividing by its magnitude gives a unit normal vector.
由于 a × b 同时垂直于 a 与 b，它是任何包含（或平行于）方向 a 与 b 的平面的法向量。再除以它的模长，就得到单位法向量。
n̂ = (a × b) ÷ |a × b|
This is the standard method for finding the unit normal to a plane given two direction vectors in the plane, and it is also the key step in finding the perpendicular distance from a point to a plane or to a line.
这是已知平面内两个方向向量时求单位法向量的标准方法，也是求点到平面或点到直线垂
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