📚 IGCSE Maths: Vectors – Essential Revision | IGCSE 数学：向量 考点精讲
Vectors are a fundamental part of the IGCSE Maths syllabus, bridging geometry and algebra. They describe quantities that have both magnitude and direction, unlike scalars which have only magnitude. Mastering vector notation, operations, and geometric applications is essential for success in both the core and extended papers. This guide walks through every key concept, from representing a vector as a column to constructing vector proofs for collinearity and parallel lines.
向量是 IGCSE 数学课程中连接几何与代数的核心内容。它们描述既有大小又有方向的量，这一点与仅有大小的标量截然不同。掌握向量的表示法、运算法则及其在几何中的应用，是通过核心卷与拓展卷的关键。本文全面梳理考点，从列向量表示到利用向量证明共线与平行，助你彻底吃透每一个要点。
1. What is a Vector? | 什么是向量？
A vector is a quantity that has both magnitude (size) and direction. Typical examples include displacement, velocity, and force. In contrast, a scalar has only magnitude, such as distance, speed, or mass. In IGCSE mathematics, vectors are often used to describe translations and to prove geometric relationships without relying on coordinate geometry.
向量是既有大小又有方向的量。常见的例子包括位移、速度和力。相比之下，标量只有大小，如距离、速率或质量。在 IGCSE 数学中，向量常用来描述平移，并在不依赖坐标几何的情况下证明几何关系。
You can think of a vector as an arrow: the length of the arrow represents its magnitude, and the arrowhead points in the vector's direction. Two vectors are equal if they have the same magnitude and the same direction, regardless of their starting points.
你可以将向量想象为一支箭：箭的长度表示其大小，箭头指向表示方向。两个向量只要大小相等、方向相同，无论起点在哪里，它们就是相等的。
2. Vector Notation | 向量的表示法
In IGCSE, a vector can be denoted in several ways. A typical representation uses a bold lowercase letter, such as a, or an underlined letter, often written as a in handwriting. When referring to a vector between two points, say from A to B, we write AB with an arrow above in textbooks, but in typed form we often use AB or simply AB with an arrow symbol. Here we will use bold: a or AB.
在 IGCSE 中，向量有多种表示方法。常见的是用小写粗体字母表示，如 a，或在手写时加下划线，如 a。当表示两点间的向量时，例如从 A 到 B，我们写作 AB。在本文中，我们用粗体来表示向量：a 或 AB。
A vector can also be written as a column vector, which is very useful for calculations. The column vector v = (
| x |
| y |
向量也可以写成列向量的形式，这对计算非常方便。列向量 v = (
| x |
| y |
3. Column Vectors | 列向量
Column vectors are the standard way to express vectors in component form. A vector a = (
| 3 |
| 4 |
| -2 |
| 5 |
列向量是以分量形式表达向量的标准方式。向量 a = (
| 3 |
| 4 |
| -2 |
| 5 |
Column vectors are added or subtracted by combining the corresponding components. This makes algebraic manipulation straightforward and links directly to transformations on the coordinate grid.
通过对应分量相加或相减，可以对列向量进行加减运算。这使代数运算变得直接，并与坐标网格上的变换紧密相连。
4. Addition and Subtraction of Vectors | 向量的加减法
To add two vectors, you add their components separately. If a = (
| a₁ |
| a₂ |
| b₁ |
| b₂ |
| a₁ + b₁ |
| a₂ + b₂ |
| a₁ - b₁ |
| a₂ - b₂ |
进行向量加法时，只需将对应分量相加。若 a = (
| a₁ |
| a₂ |
| b₁ |
| b₂ |
| a₁ + b₁ |
| a₂ + b₂ |
| a₁ - b₁ |
| a₂ - b₂ |
Geometrically, vector addition follows the triangle law: place the tail of b at the head of a; the sum a + b is the vector from the tail of a to the head of b. The parallelogram law is another visualisation. For subtraction, a - b is the same as a + (-b), where -b has the same magnitude as b but opposite direction.
在几何上，向量加法遵循三角形法则：将 b 的尾部接在 a 的头部，和向量 a + b 就是从 a 的尾部指向 b 的头部的向量。平行四边形法则提供了另一种直观理解。对于减法，a - b 等同于 a + (-b)，其中 -b 与 b 大小相等但方向相反。
5. Scalar Multiplication | 标量乘法
Multiplying a vector by a scalar (a real number) changes its magnitude but not its direction, unless the scalar is negative, in which case the direction is reversed. For a vector v = (
| x |
| y |
| kx |
| ky |
将一个向量乘以一个标量（实数）会改变其大小，但方向不变，除非标量为负数，此时方向反转。对于向量 v = (
| x |
| y |
| kx |
| ky |
posite direction. The zero vector, obtained when k = 0, has no direction and is written as 0 or (0,0) as a column.
若 k > 0，向量被拉长或缩短，方向保持不变。若 k < 0，向量指向反方向。当 k = 0 时得到零向量，它没有方向，列向量形式为 (0,0)。
6. Magnitude of a Vector | 向量的模
The magnitude (or length) of a vector v = (
| x |
| y |
| 3 |
| 4 |
向量 v = (
| x |
| y |
| 3 |
| 4 |
Remember that magnitude is always a non-negative scalar. It is often used to compare vector lengths or to find unit vectors, but unit vectors are only required in extended IGCSE.
牢记模始终是非负的标量。它常用来比较向量的长度或求单位向量，不过单位向量仅在 IGCSE 拓展卷中要求掌握。
7. Parallel Vectors | 平行向量
Two vectors are parallel if one is a scalar multiple of the other. That is, a is parallel to b if a = kb for some scalar k. This also means that if two vectors are parallel, their directions are either exactly the same (k > 0) or opposite (k < 0).
若一个向量是另一个向量的标量倍数，则这两个向量平行。也就是说，如果存在某个标量 k，使得 a = kb，则 a 与 b 平行。这也意味着，两个平行向量的方向要么完全相同 (k > 0)，要么完全相反 (k < 0)。
In column form, parallel vectors have proportional components. For instance, (
| 6 |
| 9 |
| 2 |
| 3 |
在列向量形式下，平行向量的各个分量成比例。例如，(
| 6 |
| 9 |
| 2 |
| 3 |
8. Position Vectors | 位置向量
The position vector of a point A relative to a fixed origin O is the vector OA. It gives the displacement from O to A. If A has coordinates (x, y), then its position vector is (
| x |
| y |
点 A 相对于固定原点 O 的位置向量记作 OA，它给出从 O 到 A 的位移。若点 A 的坐标为 (x, y)，则其位置向量为 (
| x |
| y |
Any vector AB can be expressed in terms of position vectors: AB = OB - OA. This relation is fundamental for vector geometry and proofs involving midpoints and ratios.
任意向量 AB 都可以用位置向量表示：AB = OB - OA。这一关系式是进行向量几何运算以及证明中点、比例问题的基础。
9. Vector Geometry – Finding Paths | 向量几何 – 路径表示
In vector geometry, you are often asked to express one vector in terms of others using a given diagram. The key strategy is to find a path from the starting point to the ending point of the desired vector, writing each segment as a known vector or combination of known vectors. For example, if you know OA and OB, then AB = OB - OA.
在向量几何中，通常需要根据给定的图将某向量用其他向量表示出来。关键策略是找到从所求向量的起点到终点的一条路径，并将每一段写为已知向量或已知向量的组合。例如，若已知 OA 和 OB，则 AB = OB - OA。
When a point divides a segment in a given ratio, you can use scalar multiples. If M is the midpoint of AB, then OM = OA + ½AB or directly OM = ½(OA + OB). More generally, if point P divides AB so that AP : PB = m : n, then OP = (n/(m+n))OA + (m/(m+n))OB.
当点按给定比例分割线段时，可运用标量倍数。若 M 是 AB 的中点，则 OM = OA + ½AB，或直接写作 OM = ½(OA + OB)。更一般地，若点 P 分 AB 使得 AP : PB = m : n，则 OP = (n/(m+n))OA + (m/(m+n))OB。
10. Vector Proofs | 向量证明
IGCSE extended
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