📚 Vector Products in Component Form | 分量形式下的向量积
The vector product, also known as the cross product, is a binary operation on two vectors in three-dimensional space. It returns a third vector that is perpendicular to the plane containing the original two, and its magnitude is proportional to the area of the parallelogram spanned by them. In AQA A-level Further Mathematics, mastering the component form of the vector product is essential for solving problems involving areas, normals and 3D geometry.
向量积（又称叉积）是三维空间中两个向量之间的一种二元运算。它产生一个同时垂直于原来两个向量所在平面的新向量，其大小与由这两个向量构成的平行四边形面积成正比。在 AQA 进阶数学考试中，熟练掌握向量积的分量形式对于求解面积、法向量以及三维几何问题至关重要。
1. Definition and Notation | 定义与记号
For two vectors a and b, the vector product is written as a × b. It is defined by two pieces of information: its magnitude and its direction.
对于两个向量 a 和 b，向量积记作 a × b。它由两个要素来定义：大小和方向。
The magnitude is given by the formula |a × b| = |a||b| sin θ, where θ is the angle between the two vectors.
其大小由公式 |a × b| = |a||b| sin θ 给出，其中 θ 是两个向量之间的夹角。
The direction of a × b is perpendicular to both a and b, determined by the right-hand rule. The result is therefore a vector, not a scalar.
a × b 的方向同时垂直于 a 和 b，由右手定则确定。因此结果是向量，而不是标量。
2. Component Form | 分量形式
Let a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k. Then the vector product in component form is:
设 a = a₁i + a₂j + a₃k，b = b₁i + b₂j + b₃k。则向量积的分量形式为：
a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k
This expression can also be remembered using a determinant with the unit vectors in the first row:
这个表达式也可以用第一行为单位向量的行列式来记忆：
a × b = i(a₂b₃ − a₃b₂) − j(a₁b₃ − a₃b₁) + k(a₁b₂ − a₂b₁)
Notice that the middle component gives a₃b₁ − a₁b₃ because of the negative sign in the expansion.
注意中间分量为 a₃b₁ − a₁b₃，这是因为展开式中含有负号。
3. Key Properties | 关键性质
The vector product obeys several important algebraic properties that are frequently tested.
向量积满足若干重要的代数性质，这些性质在考试中经常出现。
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Anti-commutativity: a × b = −(b × a).
反交换律：a × b = −(b × a)。
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Distributivity: a × (b + c) = a × b + a × c.
分配律：a × (b + c) = a × b + a × c。
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Scalar multiplication: (λa) × b = a × (λb) = λ(a × b).
标量倍数：(λa) × b = a × (λb) = λ(a × b)。
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Self-product: a × a = 0.
自身向量积：a × a = 0。
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Parallel vectors: If a and b are non-zero and a × b = 0, then a and b are parallel.
平行向量：若 a 和 b 均为非零向量且 a × b = 0，则 a 与 b 平行。
4. Geometric Interpretation | 几何意义
The magnitude of the vector product gives the area of a parallelogram.
向量积的大小给出平行四边形的面积。
Area of parallelogram = |a × b|
平行四边形面积 = |a × b|
If the two vectors represent adjacent sides of a triangle, then the area of that triangle is half the magnitude of the cross product:
如果两个向量表示三角形的两条邻边，则该三角形的面积等于向量积大小的一半：
Area of triangle = ½ |a × b|
三角形面积 = ½ |a × b|
Additionally, the direction of a × b is a normal vector to the plane containing a and b.
此外，a × b 的方向就是包含 a 和 b 的平面的一个法向量。
5. Right-Hand Rule and Unit Vector Products | 右手定则与单位向量积
The right-hand rule determines the orientation of the cross product. Curl your fingers from the first vector toward the second; your thumb points in the direction of the result.
右手定则用于确定向量积的方向。将手指从第一个向量弯向第二个向量，拇指所指即为结果的方向。
For the standard unit vectors, the cyclic products are:
对于标准单位向量，循环向量积为：
i × j = k, j × k = i, k × i = j
Reversing the order introduces a negative sign:
交换顺序会引入负号：
j × i = −k, k × j = −i, i × k = −j
| i | j | k | |
| i | 0 | k | −j |
| j | −k | 0 | i |
| k | j | −i | 0 |
6. Scalar Triple Product | 三重标量积
The scalar triple product a · (b × c) combines the dot and cross products to produce a scalar.
三重标量积 a · (b × c) 将点积和向量积结合起来，得到一个标量。
Its absolute value equals the volume of the parallelepiped formed by the three vectors:
其绝对值等于由这三个向量构成的平行六面体的体积：
Volume = |a · (b × c)|
体积 = |a · (b × c)|
In component form, this equals the determinant of the 3×3 matrix whose rows are the components of a, b, and c.
在分量形式下，它等于以 a、b、c 的坐标为行向量的 3×3 矩阵的行列式。
If a · (b × c) = 0, the three vectors are coplanar.
如果 a · (b × c) = 0，则这三个向量共面。
7. Application: Normals and Plane Equations | 应用：法向量与平面方程
A common use of the vector product is to find a normal vector to a plane. If two direction vectors d₁ and d₂ lie in the plane, then d₁ × d₂ gives a vector perpendicular to the plane.
向量积的一个常见用途是求平面的法向量。如果两个方向向量 d₁ 和 d₂ 位于平面内，则 d₁ × d₂ 给出垂直于该平面的向量。
Once the normal n is known, the equation of the plane can be written in scalar product form:
一旦求得法向量 n，平面的方程可以写成标量积形式：
r · n = a · n
where a is the position vector of a known point on the plane.
其中 a 是平面上已知点的位置向量。
This technique is often required in AQA questions that ask for the angle between planes or the line of intersection.
这种技巧在 AQA 试题中经常用于求平面间夹角或两平面交线。
8. Worked Example 1: Computing a Vector Product | 例题 1：计算向量积
Given a = 2i − j + 3k and b = i + 2j + k, find a × b.
已知 a = 2i − j + 3k，b = i + 2j + k，求 a × b。
Using the component formula: a₁ = 2, a₂ = −1, a₃ = 3; b₁ = 1, b₂ = 2, b₃ = 1.
使用分量公式：a₁ = 2，a₂ = −1，a₃ = 3；b₁ = 1，b₂ = 2，b₃ = 1。
i component: a₂b₃ − a₃b₂ = (−1)(1) − (3)(2) = −1 − 6 = −7
j component: a₃b₁ − a₁b₃ = (3)(1) − (2)(1) = 3 − 2 = 1
k component: a₁b₂ − a₂b₁ = (2)(2) − (−1)(1) = 4 + 1 = 5
Therefore a × b = −7i + j + 5k.
因此 a × b = −7i + j + 5k。
To verify perpendicularity, check the dot product with a: (2)(−7) + (−1)(1) + (3)(5) = −14 − 1 + 15 = 0. Similarly with b: (1)(−7) + (2)(1) + (1)(5) = 0.
为验证垂直性，检查与 a 的点积：(2)(−7) + (−1)(1) + (3)(5) = −14 − 1 + 15 = 0。与 b 的点积也同理：(1)(−7) + (2)(1) + (1)(5) = 0。
9. Worked Example 2: Area of a Triangle and a Unit Normal | 例题 2：三角形面积与单位法向量
Find the area of the triangle with vertices P(1,0,0), Q(0,2,0), R(0,0,3).
求顶点为 P(1,0,0)、Q(0,2,0)、R(0,0,3) 的三角形面积。
Let a = PQ = −i + 2j, and b = PR = −i + 3k.
设 a = PQ = −i + 2j，b = PR = −i + 3k。
Compute the cross product:
计算向量积：
a = (−1, 2, 0), b = (−1, 0, 3).
a = (−1, 2, 0)，b = (−1, 0, 3)。
i: (2)(3) − (0)(0) = 6
j: (0)(−1) − (−1)(3) = 3
k: (−1)(0) − (2)(−1) = 2
So a × b = 6i + 3j + 2k.
所以 a × b = 6i + 3j + 2k。
Its magnitude is √(36 + 9 + 4) = √49 = 7.
其大小为 √(36 + 9 + 4) = √49 = 7。
Therefore the area of the triangle is ½ × 7 = 3.5 square units.
因此三角形面积为 ½ × 7 = 3.5 平方单位。
A unit normal to the plane containing the triangle is (6/7)i + (3/7)j + (2/7)k.
包含该三角形的平面的单位法向量为 (6/7)i + (3/7)j + (2/7)k。
10. Common Pitfalls and Exam Tips | 常见错误与考试技巧
AQA examiners report that misordering the components and sign errors are the most frequent mistakes in vector product questions.
AQA 考官报告指出，分量顺序错误和正负号错误是向量积题目中最常见的失误。
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Always write the formula before substituting values; this helps you track each component.
代入数值前先写出公式，这有助于跟踪每个分量。
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Remember that a × b is not the same as b × a: the result is exactly the negative.
记住 a ×
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