📚 The Vector Product | 向量积
In A-Level Mathematics, the vector product (also called the cross product) is a key operation on two vectors in three-dimensional space. Unlike the scalar product, which produces a scalar, the vector product returns a new vector that is perpendicular to both original vectors. It is essential for finding areas, testing parallelism, and determining normals to planes.
在 A-Level 数学中，向量积（又称叉积）是三维空间中两个向量之间的一种重要运算。与得到标量的标量积不同，向量积得到一个新向量，该向量同时垂直于原来的两个向量。它在求面积、判断平行以及确定平面法向量等问题中非常关键。
1. Definition of the Vector Product | 向量积的定义
For two vectors a and b, with angle θ between them (0° ≤ θ ≤ 180°), the vector product is defined as:
a × b = |a||b| sin θ n̂
对于两个向量 a 和 b，它们之间的夹角为 θ（0° ≤ θ ≤ 180°），向量积定义为：
a × b = |a||b| sin θ n̂
Here n̂ is a unit vector perpendicular to both a and b, chosen so that a, b and n̂ form a right-handed system. The magnitude of the vector product is |a × b| = |a||b| sin θ. This definition only works in three dimensions.
其中 n̂ 是同时垂直于 a 和 b 的单位向量，其方向使得 a、b 与 n̂ 构成右手系。向量积的大小为 |a × b| = |a||b| sin θ。这个定义仅在三维空间中成立。
2. Right-Hand Rule and Direction | 右手定则与方向
The direction of a × b is determined by the right-hand rule. Point the fingers of your right hand in the direction of a, then curl them towards b through the smaller angle. Your thumb points in the direction of a × b. Because of this rule, a × b and b × a point in opposite directions.
a × b 的方向由右手定则确定。将右手四指指向 a 的方向，然后通过较小角度向 b 弯曲四指，拇指所指的方向就是 a × b 的方向。由于这个规则，a × b 与 b × a 指向相反的方向。
3. Component Form and Determinant Formula | 分量形式与行列式公式
If a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, then:
a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k
如果 a = a₁i + a₂j + a₃k 且 b = b₁i + b₂j + b₃k，则：
a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k
This result can be remembered using a 3 × 3 determinant with i, j, k in the first row, the components of a in the second row, and the components of b in the third row. Be especially careful with the sign of the j component: it is negative in the expansion.
这个结果可以通过一个 3 × 3 行列式来记忆：第一行是 i、j、k，第二行是 a 的分量，第三行是 b 的分量。展开时要特别注意 j 分量的符号为负。
4. Properties of the Vector Product | 向量积的性质
The vector product has the following important algebraic properties. It is anti-commutative, so a × b = −b × a. It is distributive over vector addition, so a × (b + c) = a × b + a × c. For any scalar λ, (λa) × b = λ(a × b). If a and b are parallel, then a × b = 0. If a and b are perpendicular, then |a × b| = |a||b|.
向量积具有以下重要代数性质。它满足反交换律：a × b = −b × a。它对向量加法满足分配律：a × (b + c) = a × b + a × c。对任意标量 λ，有 (λa) × b = λ(a × b)。若 a 与 b 平行，则 a × b = 0。若 a 与 b 垂直，则 |a × b| = |a||b|。
5. Relation to Area and Sine | 与面积和正弦的关系
The magnitude |a × b| = |a||b| sin θ is exactly the area of the parallelogram spanned by a and b. This gives a practical formula: area = |a × b|. For a triangle with two side vectors a and b, the area is ½ |aPublished by TutorHao | A-Level Mathematics Revision Series | aleveler.com
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