📚 Applications of Vector Products: Areas, Angles and Volumes in 3D | 向量积的应用：面积、角度与三维体积
In AQA A-Level Mathematics, vector products appear in two closely related forms: the scalar product, also called the dot product, and the vector product, also called the cross product. Both convert abstract 3D vectors into practical tools for measuring angles, areas, distances and volumes. This revision guide reviews the key definitions and then works through their main geometric and mechanical applications.
在 AQA A-Level 数学中，向量积以两种密切相关的形式出现：标量积（又称点积）和向量积（又称叉积）。两者都将抽象的三维向量转化为测量角度、面积、距离和体积的实用工具。本复习指南回顾关键定义，并逐一讲解它们在几何和力学中的主要应用。
1. Scalar Product: Definition and Core Formula | 标量积：定义与核心公式
For vectors a and b, the scalar product is written as a · b. It is defined by the cosine formula:
对于向量 a 和 b，标量积写作 a · b。它由余弦公式定义：
a · b = |a||b| cos θ
₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, then the scalar product can also be calculated as:
此处 θ 是两个向量尾尾相接时的夹角。在分量形式下，若 a = a₁i + a₂j + a₃k，b = b₁i + b₂j + b₃k，则标量积也可以计算为：
a · b = a₁b₁ + a₂b₂ + a₃b₃
The result is a scalar, not a vector, which is why the operation is called the scalar product. It is most useful when you need to find an angle, test perpendicularity, or project one vector onto another.
结果是一个标量而不是向量，因此该运算被称为标量积。当你需要求角度、判断垂直性或把一个向量投影到另一个向量上时，它最为有用。
2. Finding the Angle Between Two Vectors | 求两个向量之间的夹角
Rearranging the defining formula gives:
将定义公式变形可得：
cos θ = (a · b) / (|a||b|)
three dimensions. For example, if a = 3i + 4j and b = i + 2j, then a · b = 3×1 + 4×2 = 11, |a| = 5 and |b| = √5, so θ = cos⁻¹(11/(5√5)).
该公式在二维和三维中均适用。例如，若 a = 3i + 4j，b = i + 2j，则 a · b = 3×1 + 4×2 = 11，|a| = 5，|b| = √5，因此 θ = cos⁻¹(11/(5√5))。
This application appears frequently in AQA exam questions on straight lines and planes, especially when finding the acute angle between two direction vectors. Always take the acute angle when the question asks for the angle between lines.
这一应用经常出现在 AQA 关于直线和平面的考题中，尤其是求两个方向向量之间的锐角时。当题目要求两条直线的夹角时，始终取锐角。
3. Testing for Perpendicular Vectors | 检验垂直向量
Two non-zero vectors a and b are perpendicular if and only if their scalar product is zero:
两个非零向量 a 和 b 垂直当且仅当其标量积为零：
a · b = 0
− 3j + k and b = i + 2j + 4k are perpendicular because 2×1 + (−3)×2 + 1×4 = 2 − 6 + 4 = 0.
这是因为 cos 90° = 0，因此点积为三维几何中的直角提供了一个快速的代数检验方法。例如，a = 2i − 3j + k 与 b = i + 2j + 4k 垂直，因为 2×1 + (−3)×2 + 1×4 = 2 − 6 + 4 = 0。
This idea is also used to find a perpendicular vector
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
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