📚 Vectors | 矢量
Vectors are one of the most visual and application‑rich topics in IGCSE Edexcel Mathematics. They provide a powerful language for describing translations, forces, and geometric relationships. Mastering vector addition, scalar multiplication, and column‑vector reasoning not only secures marks in paper 2 and 4 but also builds a bridge towards A‑Level mechanics and further pure mathematics.
矢量是 IGCSE Edexcel 数学中最具直观性和应用价值的主题之一，它为描述平移、力以及几何关系提供了一种强大的语言。掌握矢量加法、标量乘法以及列矢量的推理方法，不仅能在试卷二和试卷四中稳稳拿分，还能为 A‑Level 的力学和进阶纯数学搭建桥梁。
1. What is a Vector? | 什么是矢量？
A vector is a quantity that has both magnitude (size) and direction. In contrast, a scalar has only magnitude. Displacement, velocity, and force are vector quantities; distance, speed, and mass are scalars. In IGCSE problems, a vector is often shown as a directed line segment with an arrow.
矢量是既有大小又有方向的量。标量则只有大小。位移、速度和力是矢量；路程、速率和质量是标量。在 IGCSE 题目中，矢量通常用一个带箭头的有向线段表示。
2. Notation and Representation | 符号与表示
A vector can be written in bold, such as a, or as AB with an arrow above it. In print, we use bold letters; in handwriting, a wavy underline or an arrow is common. A vector can also be expressed as a column vector, for example v = ( x / y ) — ertical pair (x, y) superscript above each other. For clarity, I will use column notation: a = (3, 4) meaning 3 units to the right and 4 units up.
矢量可以写成粗体，如 a，或带箭头的 AB。印刷体中常用粗体字母，手写时则通常加上波浪线或箭头。矢量也可用列矢量表示，例如 v 写作 (x|y)（上下排列）。为清晰起见，我将使用 a = (3, 4) 表示向右 3 个单位、向上 4 个单位。
3. Magnitude of a Vector | 矢量的大小（模）
The magnitude of a vector a = (x, y) is found using Pythagoras' theorem. It is denoted by |a| and calculated as √x² + y². For example, if a = (3, 4), then |a| = √3² + 4² = √25 = 5. The magnitude is always a non‑negative scalar.
矢量 a = (x, y) 的模通过勾股定理求得，记作 |a|，计算公式为 √x² + y²。例如，若 a = (3, 4)，则 |a| = √3² + 4² = √25 = 5。矢量的大小永远是一个非负标量。
4. Equality of Vectors | 相等的矢量
Two vectors are equal if they have the same magnitude and the same direction. Their starting points do not matter; equal vectors are parallel and of equal length. In column form, vectors a = (x₁, y₁) and b = (x₂, y₂) are equal if and only if x₁ = x₂ and y₁ = y₂.
两个矢量若大小相同、方向相同，则它们相等，与起点位置无关。相等的矢量彼此平行且长度相等。在列矢量形式下，a = (x₁, y₁) 与 b = (x₂, y₂) 相等，当且仅当 x₁ = x₂ 且 y₁ = y₂。
5. Addition of Vectors | 矢量加法
To add two vectors, we place them head‑to‑tail. The resultant vector goes from the start of the first to the end of the second. In column form, add the corresponding components: if a = (x₁, y₁) and b = (x₂, y₂), then a + b = (x₁ + x₂, y₁ + y₂). This is commutative: a + b = b + a.
矢量加法采用首尾相接的方式：将第二个矢量的起点置于第一个矢量的终点，合矢量由第一个矢量的起点指向第二个矢量的终点。在列矢量形式下，对应分量相加：若 a = (x₁, y₁)，b = (x₂, y₂)，则 a + b = (x₁ + x₂, y₁ + y₂)。矢量加法满足交换律：a + b = b + a。
6. Subtraction of Vectors | 矢量减法
Subtracting a vector is equivalent to adding its negative. The negative of v has the same magnitude but opposite direction. Thus a − b = a + (−b). In column form, subtract component‑wise: a − b = (x₁ − x₂, y₁ − y₂). Geometrically, the vector from point A to point B can be written as OB − OA.
矢量减法等价于加上其负矢量。负矢量的大小相同、方向相反。因此 a − b = a + (−b)。在列矢量形式下，对应分量相减：a − b = (x₁ − x₂, y₁ − y₂)。在几何中，从点 A 到点 B 的矢量可表示为 OB − OA。
7. Multiplication by a Scalar | 标量乘法
When a vector is multiplied by a scalar k, its magnitude is scaled by |k|, and its direction remains the same if k > 0, or reverses if k < 0. For a = (x, y), ka If k = ½, it halves in length. Scalar multiples of a vector are parallel to the original.
矢量乘以标量 k 时，其模变为原来的 |k| 倍；若 k > 0，方向保持不变；若 k < 0，方向反向。对于 a = (x, y)，有 ka = (kx, ky)。当 k = 2 时，矢量长度变为两倍；k = ½ 时长度减半。标量乘法得到的矢量与原矢量平行。
8. Column Vectors and Component Form | 列矢量与分量形式
A vector in the xy‑plane is often written as a column vector p = (pₓ, py). The top number is the horizontal displacement (x‑component), the bottom number is the vertical displacement (y‑component). For instance, u = (5, −2) means a movement of 5 right and 2 down. The vector AB with A at (2, 3) and B at (7, 1) is (7−2, 1−3) = (5, −2).
平面内的矢量常写成列矢量 p = (pₓ, py)。上方数字表示水平位移（x 分量），下方数字表示垂直位移（y 分量）。例如，u = (5, −2) 表示向右 5 个单位、向下 2 个单位。若 A 坐标为 (2, 3)、B 为 (7, 1)，则矢量 AB = (7−2, 1−3) = (5, −2)。
9. Position Vectors | 位置矢量
The position vector of a point P is the vector from the origin O to P. If P has coordinates (p, q), its position vector is OP = (p, q). The vector connecting any two points A and B is given by AB = OB − OA. This relation is extremely useful when solving geometry problems with vectors.
点 P 的位置矢量是从原点 O 到点 P 的矢量。若 P 的坐标为 (p, q)，则其位置矢量为 OP = (p, q)。任意两点 A 和 B 之间的矢量可由 AB = OB − OA 求得。这一关系在利用矢量求解几何问题时极为有用。
10. Vector Geometry — Midpoints and Ratios | 矢量几何 — 中点和比例
If M is the midpoint of AB, then OM = (OA + OB division in a ratio m : n, the point P on AB such that AP : PB = m : n has position vector OP = (nOA + mOB) / (m + n). These formulas allow you to locate points without coordinate grids, just using vector algebra.
若 M 为线段 AB 的中点，则 OM = (OA + OB) / 2。对于按比例 m : n 内分的情况，点 P 满足 AP : PB = m : n，其位置矢量为 OP = (nOA + mOB) / (m + n)。借助这些公式，无需坐标网格，仅用矢量代数就能确定点的位置。
11. Proving Collinearity and Parallelism | 证明共线和平行
Two vectors are parallel if one is a scalar multiple of the other. Three points A, B, and C are collinear if AB and AC are parallel — that is, if AB = kAC for some scalar k. In exam questions, you may be asked to show that points lie on a straight line by expressing one vector as a multiple of another.
两个矢量平行意味着其中一个可以表示为另一个的标量倍数。三点 A、B、C 共线，当且仅当 AB 与 AC 平行，即存在某个标量 k 使得 AB = kAC。考试中常要求通过将一个矢量表示为另一个矢量的倍数来证明某些点在同一直线上。
12. Typical Exam Questions and Tips | 常见考题与应对技巧
IGCSE vector questions often involve finding an unknown component, computing a magnitude, or expressing a path as a combination of given vectors. Always show clear column‑addition steps. When proving collinearity, state the scalar multiple explicitly. Drawing a simple sketch helps avoid sign errors. Common pitfalls include mixing up AB and BA: remember BA = −AB.
IGCSE 矢量题目常涉及求未知分量、计算模，或者将某一路径表示为已知矢量的组合。务必清晰地展示列矢量相加的步骤。证明共线时，要明确写出标量倍数。画一个简单的草图有助于避免符号错误。常见易错点包括混淆 AB 与 BA，切记 BA = −AB。
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