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Vector Products in Component Form | 分量形式下的向量积

2026-08-30 06:19 · 本文部分内容由AI辅助创作

📚 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。

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

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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)|

onent 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。