📚 The Vector Product | 向量积
The vector product (also called the cross product) is a binary operation on two vectors in three-dimensional space. Unlike the scalar product, which gives a scalar result, the vector product gives a vector that is perpendicular to both original vectors. This operation is essential for finding normals to planes, areas of parallelograms, and solving geometric problems in 3D Mechanics.
向量积（又称叉积）是三维空间中两个向量的二元运算。与得到标量结果的标量积不同，向量积的结果是一个向量，该向量同时垂直于原来的两个向量。这种运算对于求平面的法向量、平行四边形的面积以及解决三维力学中的几何问题至关重要。
1. Definition of the Vector Product | 向量积的定义
The vector product of two vectors a and b is denoted by a × b (read 'a cross b'). It is defined as a vector with magnitude |a||b| the angle between the vectors (0 ≤ θ ≤ π), and with direction given by the right-hand rule — perpendicular to both a and b.
两个向量 a 和 b 的向量积记为 a × b（读作“a 叉乘 b”）。其定义是：大小为 |a||b| sin θ 的向量，其中 θ 是两向量之间的夹角（0 ≤ θ ≤ π），方向由右手定则确定——垂直于 a 和 b 所在的平面。
|a × b| = |a||b| sin θ
If a = 0 or b = 0, the vector product is the zero vector. Note that the vector product is only defined for 3D vectors; in AQA A-Level Maths you will use vectors written in component form i, j, k or as column vectors.
如果 a = 0 或 b = 0，则向量积为零向量。请注意，向量积仅对三维向量有定义；在 AQA A-Level 数学中，你会使用分量形式 i, j, k 或列向量表示的向量。
2. Component Formula | 分量公式
Given a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, the vector product is computed using a 3×3 determinant:
已知 a = a₁i + a₂j + a₃k，b = b₁i + b₂j + b₃k，向量积通过以下 3×3 行列式计算：
a × b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k
This can also be written as a determinant expansion:
这也可以写成行列式展开的形式：
| a × b = det | i j k a₁ a₂ a₃ b₁ b₂ b₃ |
entries with a positive sign; for the j-component, cover the second column, compute the determinant with a negative sign; for the k-component, cover the third column, compute with a positive sign.
一个有用的记忆技巧：求 i 分量时，遮住第一列，对剩余元素求 2×2 行列式，取正号；求 j 分量时，遮住第二列，求行列式取负号；求 k 分量时，遮住第三列，求行列式取正号。
3. Worked Example 1 | 示例 1
Let a = 2i + 3j − k and b = i − 2j + 4k. Find a × b.
设 a = 2i + 3j − k，b = i − 2j + 4k。求 a × b。
Using the component formula:
利用分量公式：
a × b = (3×4 − (−1)(−2))i − (2×4 − (−1)(1))j + (2×(−2) − 3×1)k
Calculate each component:
逐项计算各分量：
- i-component: (3×4) − ((−1)×(−2)) = 12 − 2 = 10
- i 分量： (3×4) − ((−1)×(−2)) = 12 − 2 = 10
- j-component: −[(2×4) − ((−1)×1)] = −(8 − (−1)) = −9
- j 分量： −[(2×4) − ((−1)×1)] = −(8 − (−1)) = −9
- k-component: (2×(−2)) − (3×1) = −4 − 3 = −7
- k 分量： (2×(−2)) − (3×1) = −4 − 3 = −7
Therefore a × b = 10i − 9j − 7k. This is the vector perpendicular to both a and b.
因此 a × b = 10i − 9j − 7k。这就是同时垂直于 a 和 b 的向量。
4. Properties of the Vector Product | 向量积的性质
You must know the key properties, especially the anti-commutative nature, which differs from ordinary multiplication.
你必须掌握以下关键性质，尤其是反交换性，这与普通乘法不同。
- Anti-commutative: a × b = −(b × a)
- 反交换律： a × b = −(b × a)
- Parallel vectors: If a and b are parallel (θ = 0 or π), then a × b = 0
- 平行向量： 若 a 与 b 平行（θ = 0 或 π），则 a × b = 0
- Self product: a × a = 0
- 自身叉积： a × a = 0
- Scalar multiplication: (ka) × b = a × (kb) = k(a × b) for any scalar k
- 标量倍乘： 对任意标量 k，(ka) × b = a × (kb) = k(a × b)
- Distributive: a × (b + c) = a × b + a × c
- 分配律： a × (b + c) = a × b + a × c
The unit vector (i, j, k) products are worth memorising:
单位向量（i, j, k）的叉积结果值得记忆：
| × | i | j | k |
| i | 0 | k | −j |
| j | −k | 0 | i |
| k | j | −i | 0 |
le, i × j = k, j × k = i, and k × i = j. Reversing the order changes the sign: j × i = −k.
例如，i × j = k，j × k = i，k × i = j。交换顺序会改变符号：j × i = −k。
5. Geometric Interpretation: Area | 几何意义：面积
The magnitude |a × b| gives the area of the parallelogram formed by vectors a and b. The area of the triangle formed by a and b is half of this magnitude.
模长 |a × b| 给出由向量 a 和 b 构成的平行四边形的面积。由 a 和 b 构成的三角形面积为其一半。
Parallelogram area = |a × b|, Triangle area = ½ |a × b|
This is particularly useful when coordinates are given in 3D and finding the perpendicular height via trigonometry is inconvenient.
当给定 3D 坐标且用三角函数求垂直高度不便时，这一性质尤为实用。
6. Worked Example 2: Area of a Triangle | 示例 2：三角形面积
Points A(1, 0, 2), B(3, 2, −1) and C(0, 1, 4) form a triangle. Find its area.
点 A(1, 0, 2)、B(3, 2, −1) 和 C(0, 1, 4) 构成一个三角形。求其面积。
First form two vectors from A:
首先从 A 点构造两个向量：
- AB = (3 − 1)i + (2 − 0)j + (−1 − 2)k = 2i + 2j − 3k
- AB = (3 − 1)i + (2 − 0)j + (−1 − 2)k = 2i + 2j − 3k
- AC = (0 − 1)i + (1 − 0)j + (4 − 2)k = −i + j + 2k
- AC = (0 − 1)i + (1 − 0)j + (4 − 2)k = −i + j + 2k
Compute AB × AC:
计算 AB × AC：
= (2×2 − (−3)(1))i − (2×2 − (−3)(−1))j + (2×1 − 2×(−1))k
- i: 4 + 3 = 7
- j: −(4 − 3) = −1
- k: 2 + 2 = 4
So AB × AC = 7i − j + 4k. Its magnitude is √(7² + (−1)² + 4²) = √49 + 1 + 16 = √66.
因此 AB × AC = 7i − j + 4k。其模长为 √(7² + (−1)² + 4²) = √49 + 1 + 16 = √66。
Triangle area = ½ × √66 square units.
三角形面积 = ½ × √66 平方单位。
7. Normal Vector to a Plane | 平面的法向量
Given two non-parallel vectors d₁ and d₂ lying in a plane, the vector d₁ × d₂ is perpendicular to the plane, hence it is a normal vector n to the plane.
给定平面内两个不平行向量 d₁ 和 d₂，则向量 d₁ × d₂ 垂直于该平面，因此它是该平面的法向量 n。
If a plane has equation r · n = d, and point A with position vector a lies in the plane, you can find the Cartesian equation using n = d₁ × d₂, then (r − a) · n = 0.
若平面方程为 r · n = d，且点 A 的位置向量为 a 且位于该平面内，你可以用 n = d₁ × d₂ 求出笛卡尔方程，即 (r − a) · n = 0。
For AQA A-Level, you may be asked to find the equation of a plane given three points in the plane. The cross product provides the most direct method.
在 AQA A-Level 考试中，你可能会被要求根据平面上的三个点求出平面方程。向量积提供了最直接的方法。
8. Worked Example 3: Equation of a Plane | 示例 3：平面方程
Find the Cartesian equation of the plane passing through A(1, 2, 1), B(2, 0, 3) and C(0, 1, 2).
求过点 A(1, 2, 1)、B(2, 0, 3) 和 C(0, 1, 2) 的平面的笛卡尔方程。
Step 1 — find two vectors in the plane:
步骤 1——求平面内两个向量：
- AB = i − 2j + 2k
- AB = i − 2j + 2k
- AC = −i − j + k
- AC = −i − j + k
Step 2 — compute the normal vector n = AB × AC:
步骤 2——计算法向量 n = AB × AC：
= ((−2)(1) − (2)(−1))i − ((1)(1) − (2)(−1))j + ((1)(−1) − (−2)(−1))k
- i: −2 + 2 = 0
- j: −(1 + 2) = −3
- k: −1 − 2 = −3
So n = 0i − 3j − 3k, which we can simplify to n = j + k (dividing by −3). Using point A, a = i + 2j + k:
因此 n = 0i − 3j − 3k，可简化为 n = j + k（除以 −3）。使用点 A，a = i + 2j + k：
(r − a) · n = 0
+ zk. Then (x − 1)i + (y − 2)j + (z − 1)k dotted with (j + k) gives (y − 2) + (z − 1) = 0.
设 r = xi + yj + zk。则 (x − 1)i + (y − 2)j + (z − 1)k 与 (j + k) 点乘得到 (y − 2) + (z − 1) = 0。
y + z = 3
This is the required Cartesian equation of the plane.
这就是所求平面的笛卡尔方程。
9. Vector Product in Mechanics | 向量积在力学中的应用
In AQA A-Level Further Mathematics (or A-Level Maths with mechanics option), the vector product appears in the context of moments of a force about a point. The moment of a force F acting at a position vector r relative to a point P is defined as:
在 AQA A-Level 进阶数学（或 A-Level 数学的力学模块）中，向量积出现在力对点之矩的背景下。力 F 作用在相对于点 P 的位置向量 r 上时，力矩定义为：
M = r × F
The direction of M gives the axis of rotation (right-hand screw rule), and its magnitude gives the turning effect. In rigid-body problems, this vector formulation avoids the need to find perpendicular distances manually.
M 的方向给出旋转轴（右手螺旋法则），其大小给出转动效应的大小。在刚体问题中，这种向量表述避免手动求垂直距离的需要。
10. Common Mistakes and Exam Tips | 常见错误与考试提示
Students frequently lose marks on vector product questions for avoidable reasons. Here are the most common pitfalls and how to avoid them.
学生经常因可以避免的原因在向量积题目中失分。以下是最常见的陷阱及避免方法。
- Wrong sign for the j-component: e : The j-component in a × b is always subtracted. Remember: i positive, j negative, k positive.
- j 分量符号错误： a × b 的 j 分量始终为负。记住：i 正、j 负、k 正。
- Confusing a × b with b × a: They are opposites. Always check the order of the vectors in the question.
- 混淆 a × b 与 b × a： 它们是相反向量。始终检查题目中向量的顺序。
- Forgetting to simplify the normal vector: A normal vector can be multiplied by any non-zero scalar; simplifying (e.g., dividing by a common factor) makes the plane equation cleaner.
- 忘记化简法向量： 法向量可以乘以任何非零标量；化简（例如除以公因子）使平面方程更简洁。
- Using the scalar product formula accidentally: The scalar product uses cos θ; the vector product uses sin θ. Read the question carefully.
- 不小心使用标量积公式： 标量积使用 cos θ；向量积使用 sin θ。仔细阅读题目。
Examiners often award method marks even for incorrect calculations, so always write down the determinant or component formula clearly before substituting values.
考官通常会为即使计算错误的题目给方法分，因此务必在代入数值前清楚写出行列式或分量公式。
11. Practice Questions | 练习题
Test yourself with these AQA-style questions. Full solutions are provided below.
用以下 AQA 风格题目自测。完整解答附后。
Question 1: Given p = 3i − 2j + k and q = i + 4j − 2k, find p × q.
问题 1： 已知 p = 3i − 2j + k，q = i + 4j − 2k，求 p × q。
Question 2: The points P(2, −1, 3), Q(4, 1, 0), R(0, 2, 1) define a triangle. Find the area of triangle PQR correct to 3 significant figures.
问题 2： 点 P(2, −1, 3)、Q(4, 1, 0)、R(0, 2, 1) 确定一个三角形。求三角形 PQR 的面积（精确到 3 位有效数字）。
Question 3: Find a unit vector perpendicular to both u = i − j + 2k and v = 2i + 3j − k.
问题 3： 求同时垂直于 u = i − j + 2k 和 v = 2i + 3j − k 的单位向量。
12. Solutions to Practice Questions | 练习解答
Solution 1:
解答 1：
p × q = ((−2)(−2) − (1)(4))i − ((3)(−2) − (1)(1))j + ((3)(4) − (−2)(1))k
p × q = ((−2)(−2) − (1)(4))i − ((3)(−2) − (1)(1))j + ((3)(4) − (−2)(1))k
= (4 − 4)i − (−6 − 1)j + (12 + 2)k = 0i + 7j + 14k.
= (4 − 4)i − (−6 − 1)j + (12 + 2)k = 0i + 7j + 14k。
p × q = 7j + 14k
Solution 2: First PQ = 2i + 2j − 3k and PR = −2i + 3j − 2k. Then PQ × PR 2) − (−3)(−2))j + ((2)(3) − (2)(−2))k = (−4 + 9)i − (−4 − 6)j + (6 + 4)k = 5i + 10j + 10k.
解答 2： 首先 PQ = 2i + 2j − 3k，PR = −2i + 3j − 2k。则 PQ × PR −2) − (−3)(−2))j + ((2)(3) − (2)(−2))k = (−4 + 9)i − (−4 − 6)j + (6 + 4)k = 5i + 10j + 10k。
Magnitude = √5² + 10² + 10² = √225 = 15. Area = ½ × 15 = 7.5 square units.
模长 = √5² + 10² + 10² = √225 = 15。面积 = ½ × 15 = 7.5 平方单位。
Solution 3: u × v 1)(−1) − (2)(2))j + ((1)(3) − (−1)(2))k = (1 − 6)i − (−1 − 4)j + (3 + 2)k = −5i + 5j + 5k.
解答 3： u × v (1)(−1) − (2)(2))j + ((1)(3) − (−1)(2))k = (1 − 6)i − (−1 − 4)j + (3 + 2)k = −5i + 5j + 5k。
Its magnitude is √25 + 25 + 25 = √75 = 5√3. Therefore the unit vector is (−1/√3)i + (1/√3)j + (1/√3)k, i.e., dividing all components by 5√3.
其模长为 √25 + 25 + 25 = √75 = 5√3。因此单位向量为 (−1/√3)i + (1/√3)j + (1/√3)k，即所有分量除以 5√3。
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