📚 Edexcel IGCSE Maths Topic 47: Vectors | 爱德思IGCSE数学主题47：向量
In Edexcel IGCSE Mathematics, vectors provide a powerful language for describing movement, forces, and geometric relationships with both magnitude and direction. This article will guide you through the key concepts of vectors, from basic notation to solving geometric problems, helping you build the skills needed for exam success.
在爱德思IGCSE数学中，向量提供了一种强大的语言，用于描述同时具有大小和方向的运动、力以及几何关系。本文将从基本符号到几何问题的求解，带你逐步掌握向量的核心概念，为考试成功打下扎实基础。
1. Introduction to Vectors | 向量简介
A scalar quantity has only magnitude (size) and no direction. Common examples are mass (5 kg), speed (30 m/s), and distance (100 m). A vector quantity, however, possesses both magnitude and direction. Typical vector quantities include velocity (40 m/s north), displacement (5 km east), and force (10 N downwards). In written mathematics, vectors are often represented in bold type, such as a, or with an arrow above the letter like OA→. In this article we use bold characters for vectors.
标量只有大小，没有方向。常见的例子有质量（5 kg）、速率（30 m/s）和距离（100 m）。而向量既具有大小又具有方向，典型的向量包括速度（40 m/s 向北）、位移（5 km 向东）和力（10 N 向下）。在数学书写中，向量通常用粗体字母表示，例如 a，或在字母上方加箭头，如 OA→。本文中我们使用粗体字符表示向量。
2. Representing Vectors | 向量的表示
A vector in two dimensions can be broken down into horizontal and vertical components. The unit vectors i and j are defined as vectors of magnitude 1 along the positive x-axis and positive y-axis respectively. Any vector v can be written as v = xi + yj, where x is the change in the x-direction and y in the y-direction. For example, a displacement of 3 units right and 2 units up is given by v = 3i + 2j. An alternative column vector notation is frequently used in translations.
二维向量可以分解为水平分量和垂直分量。单位向量 i 和 j 分别定义为沿 x 轴正向和 y 轴正向、大小为 1 的向量。任何向量 v 都可以写成 v = xi + yj，其中 x 是 x 方向的变化量，y 是 y 方向的变化量。例如，向右 3 个单位、向上 2 个单位的位移可表示为 v = 3i + 2j。另一种列向量的记法在平移变换中常用。
3. Vector Addition and Subtraction | 向量的加法与减法
If a = x₁i + y₁j and b = x₂i + y₂j, then their sum is a + b = (x₁ + x₂)i + (y₁ + y₂)j. Geometrically, addition corresponds to placing the tail of b at the head of a; the resultant vector is the diagonal of the parallelogram formed. Subtraction a − b = a + (−b), where −b is the vector with the same magnitude but opposite direction, i.e., −x₂i − y₂j.
若 a = x₁i + y₁j，b = x₂i + y₂j，则它们的和为 a + b = (x₁ + x₂)i + (y₁ + y₂)j。在几何上，加法相当于将 b 的尾部放置于 a 的头部；合向量为由此构成的平行四边形的对角线。减法为 a − b = a + (−b)，其中 −b 是与 b 大小相同、方向相反的向量，即 −x₂i − y₂j。
4. Scalar Multiplication | 标量乘法
Multiplying a vector v = xi + yj by a scalar k gives kv = kxi + kyj. The magnitude is scaled by |k|, while the direction stays the same if k > 0 and reverses if k < 0. This property defines parallel vectors: if a = kb for some non-zero scalar k, then a and b are parallel (or collinear). For example, a = 6i − 4j and b = −3i + 2j are parallel because a = −2b.
将向量 v = xi + yj 与标量 k 相乘得到 kv = kxi + kyj。向量的大小缩放 |k| 倍，方向在 k > 0 时保持不变，在 k < 0 时反向。这一性质定义了平行向量：若对某个非零标量 k 有 a = kb，则 a 与 b 平行（或共线）。例如，a = 6i − 4j 与 b = −3i + 2j 平行，因为 a = −2b。
5. Magnitude of a Vector | 向量的模
The magnitude (length) of a vector v = xi + yj is found using Pythagoras' theorem: |v| = √x² + y². For example, |3i + 4j| = √3² + 4² = √25 unit vector is a vector with magnitude 1. To find the unit vector in the same direction as v, divide v by its magnitude: û = v / |v|. This is useful for specifying direction without changing speed or size.
向量 v = xi + yj 的模（长度）可由勾股定理求得：|v| = √x² + y²。例如，|3i + 4j| = √3² + 4² = √25 = 5。单位向量是大小为 1 的向量。要得到与 v 同方向的单位向量，可将 v 除以其模：û = v / |v|。这在仅需指明方向而不改变速度或大小的情况下非常有用。
6. Position Vectors | 位置向量
A position vector locates a point relative to the origin O. For a point A with coordinates (x, y), the position vector is a = xi + yj. If two points A and B have position vectors a and b, then the vector from A to B is AB = b − a. This is one of the most important results in vector geometry — it allows us to express any side of a shape in terms of the position vectors of its vertices.
位置向量用于确定一点相对于原点 O 的位置。若点 A 的坐标为 (x, y)，则其位置向量为 a = xi + yj。若点 A 和点 B 的位置向量分别为 a 和 b，则从 A 到 B 的向量为 AB = b − a。这是向量几何中最重要的结果之一——它使我们能够利用顶点的位置向量表示图形中的任意边。
7. Vectors in Geometry | 向量在几何中的应用
With position vectors we can prove geometric facts efficiently. To show that two vectors AB and CD are parallel, we check whether AB = kCD for some scalar k. To prove three points A, B, C are collinear, we show that AB = kBC (with B as a common point). The midpoint M of AB has position vector m = (a + b) / 2. These techniques replace lengthy angle and length reasoning with concise algebraic steps.
利用位置向量，我们可以高效地证明几何事实。要证明两向量 AB 与 CD 平行，只需验证是否存在标量 k 使得 AB = kCD。要证明三点 A、B、C 共线，可证明 AB = kBC（以 B 为公共点）。AB 的中点 M 的位置向量为 m = (a + b) / 2。这些方法用简洁的代数步骤替代了冗长的角度和长度推理。
8. Solving Geometric Problems with Vectors | 运用向量解决几何问题
Let us apply vectors to a classic triangle problem. In triangle ABC, let D be the midpoint of BC. We express AD in terms of AB and AC. Using position vectors with origin A: AB = b, AC = c, so D has position vector d = (b + c)/2. Then AD = d − 0 = (b + c)/2. Hence AD = ½ (AB + AC). This proves the median vector relationship. Similar strategies can be used to prove mid-point theorems and to find ratios along a line segment.
我们用一个经典的三角形问题来演示向量的运用。在三角形 ABC 中，设 D 为 BC 的中点。我们要用 AB 和 AC 表示 AD。以 A 为原点建立位置向量：AB = b，AC = c，则 D 的位置向量为 d = (b + c)/2。于是 AD = d − 0 = (b + c)/2。因此 AD = ½ (AB + AC)。这就证明了中线向量的关系。类似的策略可用于证明中点定理，以及在一条线段上求分点之比。
9. Vectors and Translations | 向量与平移变换
A translation is a transformation that moves each point of a shape by the same distance in the same direction. It can be described by a translation vector, often written as a column vector. In our i,j notation, translating a shape by the vector t = ai + bj means adding a to every x-coordinate and b to every y-coordinate. For example, translation by 3i − 2j shifts the point (x, y) to (x+3, y−2). Vectors allow us to describe translations without needing a grid or coordinates.
平移是一种将图形中的每个点沿相同的方向移动相同距离的变换。它可以用一个平移向量
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