📚 Vectors | 向量
A vector is a quantity that has both magnitude and direction, unlike a scalar which only has magnitude. In IGCSE Mathematics, vectors are used to describe translations, forces, velocities, and geometric relationships. Understanding vectors is essential for solving problems in coordinate geometry, mechanics, and pure mathematics. This article covers the key concepts of vectors you need for the Edexcel IGCSE exam, from basic notation to geometric applications.
向量是一种既有大小又有方向的量，区别于只有大小的标量。在 IGCSE 数学中，向量用于描述平移、力、速度以及几何关系。理解向量对于解决坐标几何、力学和纯数学问题至关重要。本文涵盖 Edexcel IGCSE 考试中向量的核心概念，从基本记法到几何应用。
1. What are Vectors? | 什么是向量？
A vector is defined by its magnitude (length) and direction. Scalars, such as distance, speed, or mass, only have size. Vectors are usually represented by a directed line segment with an arrow showing the direction. For example, a displacement of 5 metres east is a vector, while a distance of 5 metres is a scalar. In IGCSE, we work mainly with vectors in a plane (2‑dimensional).
向量由其大小（长度）和方向来定义。标量，例如距离、速率或质量，仅具有大小。向量通常用带有箭头表示方向的有向线段来表示。例如，向东 5 米的位移是一个向量，而 5 米的距离则是一个标量。在 IGCSE 中，我们主要研究平面（二维）中的向量。
2. Vector Notation and Representation | 向量的记法与表示
Vectors can be written in several ways. In print, a bold letter, such as a, is often used, while in handwriting you may underline the letter: a̲. A vector between two points A and B is written as AB with an arrow above: AB⃗. The order of letters gives the direction: AB⃗ goes from A to B, whereas BA⃗ goes from B to A, and AB⃗ = −BA⃗.
向量有多种写法。在印刷体中常使用粗体字母，如 a；手写时可在字母下方加下划线：a̲ 。点 A 到点 B 的向量表示为带箭头的 AB⃗。字母顺序指明了方向：AB⃗ 从 A 指向 B，BA⃗ 从 B 指向 A，且 AB⃗ = −BA⃗。
3. Column Vectors | 列向量
A convenient way to describe a vector in 2D is by using a column vector. The top number gives the horizontal displacement (positive to the right, negative to the left) and the bottom number gives the vertical displacement (positive up, negative down). For example, the vector v = (3, 2)ᵀ means 3 units right and 2 units up. We write this as:
v = (3 over 2) (but shown as column vector in text)
描述二维向量的一种便捷方式是使用列向量。上方的数字表示水平位移（正数向右，负数向左），下方的数字表示垂直位移（正数向上，负数向下）。例如，向量 v = (3, 2)ᵀ 表示向右 3 个单位、向上 2 个单位。我们将其写作：
v = &lgroup;³₂&rgroup;
Note: On paper, it is written as a vertical bracket. In pure text we sometimes use (3, 2) but the column form is standard. The vector (0, 0) is the zero vector, 0.
注意：在纸上写成竖排括号。纯文本中有时用 (3, 2) 表示，但列向量形式是标准的。向量 (0, 0) 是零向量，记作 0。
4. Adding and Subtracting Vectors | 向量的加减法
To add two vectors, simply add their corresponding components. For vectors a = (x₁, y₁)ᵀ and b = (x₂, y₂)ᵀ, the sum is a + b = (x₁+x₂, y₁+y₂)ᵀ. Graphically, this follows the “triangle rule”: place the tail of b at the head of a; the result goes from the tail of a to the head of b. Subtraction a − b is the same as adding the negative of b: a + (−b).
向量相加时，只需将对应分量相加。若向量 a = (x₁, y₁)ᵀ、b = (x₂, y₂)ᵀ，则和为 a + b = (x₁+x₂, y₁+y₂)ᵀ。图形上遵循“三角形法则”：将 b 的尾端放在 a 的尖端，则合向量从 a 的尾端指向 b 的尖端。减法 a − b 等同于加上 b 的负向量：a + (−b)。
5. Scalar Multiplication of Vectors | 向量的标量乘法
When a vector is multiplied by a scalar (a real number), each component is multiplied by that number. If v = (x, y)ᵀ and k is a scalar, then kv = (kx, ky)ᵀ. The magnitude changes by the factor |k| he direction stays the same if k is positive, or reverses if k is negative. For example, 2v stretches v by a factor of 2; −v gives the vector with same length but opposite direction.
向量乘以标量（实数）时，每个分量都乘以该数。若 v = (x, y)ᵀ、k 为标量，则 kv = (kx, ky)ᵀ。向量的大小按 |k| 倍变化，方向若 k 为正则不变，若 k 为负则反向。例如 2v 将 v 伸长为原来的 2 倍；−v 给出长度相同但方向相反的向量。
6. Magnitude of a Vector | 向量的模
The magnitude (or length) of a vector v = (x, y)ᵀ is calculated using Pythagoras’ theorem:
|v| = √x² + y²
For example, the magnitude of (3, 4)ᵀ is √3² + 4² = √25 = 5. The magnitude is always non‑negative. A unit vector has magnitude 1. To find a unit vector in the direction of v, divide v by its magnitude: û = v / |v|.
向量 v = (x, y)ᵀ 的模（或长度）使用勾股定理计算：|v| = √x² + y²。例如 (3, 4)ᵀ 的模为 √3² + 4² = √25 = 5。模总是非负的。单位向量是模为 1 的向量。要求 v 方向的单位向量，用 v 除以其模：û = v / |v|。
7. Parallel Vectors | 平行向量
Two vectors a and b are parallel if one is a scalar multiple of the other, i.e. a = kb ectors are parallel, see if the ratios of their components are equal. For instance, (2, 4)ᵀ is parallel to (1, 2)ᵀ because (2, 4)ᵀ = 2 × (1, 2)ᵀ. In geometry, this is used to prove that line segments are parallel.
如果两个向量中一个是另一个的标量倍数，即 a = k·b，则它们是平行的。平行向量方向相同或相反。要判断两个列向量是否平行，可检查它们对应分量的比值是否相等。例如 (2, 4)ᵀ 与 (1, 2)ᵀ 平行，因为 (2, 4)ᵀ = 2 × (1, 2)ᵀ。在几何中，这用来证明线段平行。
8. Position Vectors | 位置向量
A position vector of a point A is the vector from the origin O to A. If A has coordinates (x, y), then the position vector OA⃗ = (x, y)ᵀ. The vector AB⃗ can be expressed using position vectors: AB⃗ = OB⃗ − OA⃗. This relation is extremely useful in solving problems that involve moving from one point to another via the origin.
点 A 的位置向量是从原点 O 指向 A 的向量。若 A 的坐标为 (x, y)，则位置向量 OA⃗ = (x, y)ᵀ。向量 AB⃗ 可以用位置向量表示：AB⃗ = OB⃗ − OA⃗。这一关系在解决涉及经过原点的两点间向量问题时非常有用。
9. Vector Geometry – Midpoints and Ratios | 向量几何 – 中点与比例
Vectors greatly simplify geometric results. The midpoint M of line segment AB has position vector OM⃗ = ½ (OA⃗ + OB⃗). If a point P divides AB in the ratio λ:μ, then OP⃗ = (μ OA⃗ + λ OB⃗) / (λ+μ). When working without coordinates, you can use known parallel vectors and scalar multiples to find unknown vectors on a line.
向量大大简化了几何结论。线段 AB 的中点 M 的位置向量是 OM⃗ = ½ (OA⃗ + OB⃗)。若点 P 将 AB 按比例 λ:μ 分割，则 OP⃗ = (μ OA⃗ + λ OB⃗) / (λ+μ)。在没有坐标的情况下，可以利用已知的平行向量和标量倍数来求直线上未知的向量。
10. Solving Geometry Problems with Vectors | 运用向量解决几何问题
Typical IGCSE vector problems ask you to find an expression for a vector in terms of given vectors, prove that points are collinear, or show that segments are parallel. The general strategy is:
- Identify a convenient origin and express given points as position vectors.
- Use vector addition, subtraction, and scalar multiplication to build paths between points.
- For collinearity, show that one vector is a scalar multiple of another and that they share a common point.
- Always simplify your answers and use correct vector notation.
典型的 IGCSE 向量问题要求你用给定向量表示出某个向量，证明三点共线或证明线段平行。常规策略如下：
- 选择一个方便的原点，将已知点表示为位置向量。
- 使用向量加法、减法和标量乘法构建点与点之间的路径。
- 共线时，需证明一个向量是另一个的标量倍数，且它们有一个公共点。
- 始终化简答案，并使用正确的向量记法。
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