Proof of the Distributive Law | 分配律的证明

📚 Proof of the Distributive Law | 分配律的证明

a · b + a · c, from the component definition of the dot product, and explore the same law in geometry, matrices and sets.

在 A-Level 数学的核心，有一小组使代数成立的构造性性质。分配律就是允许一种运算穿过另一种运算的规则：a(b + c) = ab + ac。本文将根据点积的分量定义证明其向量形式 a · (b + c) = a · b + a · c，并在几何、矩阵与集合中考察这一定律。

1. What Is the Distributive Law? | 什么是分配律？

single property that connects multiplication with addition; without it, expanding brackets and factorising would be impossible.

实数域上的分配律指出 a(b + c) = ab + ac。它是把乘法与加法联系起来的唯一性质；没有它，去括号与因式分解都将无法进行。

For vectors there are two distributive laws, both of which can be proved from the component definition of a vector:

对向量而言有两条分配律，二者都可以从向量的分量定义出发加以证明：

• b) = ma + mb, where m is a scalar: scalar multiplication distributes over vector addition.

  m(a + b) = ma + mb，其中 m 为标量：数乘对向量加法分配。

• a · (b + c) = a · b + a · c: the dot product distributes over vector addition.

  a · (b + c) = a · b + a · c：点积对向量加法分配。

The second law is the focus of this article because it is the one most often probed in AQA examination questions that request a formal proof. The first law is proved in Section 5.

本文以第二条定律为重点，因为它是 AQA 考题中最常要求正式证明的内容。第一条定律将在第 5 节中证明。

An important remark: for real numbers these laws are axioms and need no proof, but for vectors they must be established, because vectors are defined through components and a dot product is a new operation with its own definition.

一个重要说明：对实数而言这些定律是公理，无需证明；但对向量而言必须加以确立，因为向量是通过分量定义的，而点积是一种具有自身定义的新运算。

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2. Statement of the Vector Distributive Law | 向量分配律的表述

Let a, b and c be vectors in three-dimensional space, written in terms of the standard basis vectors i, j and k:

设 a、b、c 为三维空间中的向量，用标准基向量 i、j、k 表示为：

a = a₁i + a₂j + a₃k, b = b₁i + b₂j + b₃k, c = c₁i + c₂j + c₃k

where a₁, a₂, a₃, b₁, b₂, b₃, c₁, c₂, c₃ are real numbers.

其中 a₁, a₂, a₃, b₁, b₂, b₃, c₁, c₂, c₃ 均为实数。

Theorem (distributive law for the dot product):

定理（点积分配律）：

a · (b + c) = a · b + a · c

the symmetric form (a + b) · c = a · c + b · c follows immediately from the theorem above.

又因点积满足交换律 a · b = b · a，对称形式 (a + b) · c = a · c + b · c 立即可由上定理推出。

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3. Component Proof | 分量证明

This is the cleanest and most rigorous method. It is a direct proof: we start from the definition of the dot product and manipulate symbols until the conclusion appears.

这是最简洁、最严谨的方法。它属于直接证明：从点积的定义出发，对符号进行运算直到结论出现。

Step 1. Add b and c component-wise.

步骤 1：对 b 与

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