📚 GCSE OCR Maths: Vectors – Key Exam Points | GCSE OCR 数学：向量考点精讲
Vectors are a fundamental topic in the GCSE OCR Mathematics syllabus, appearing in both Foundation and Higher tiers. Understanding vectors is essential for solving geometric problems, proving collinearity, and working with translations. This guide breaks down every key concept, from notation to vector proofs, with clear examples and exam tips to help you master vectors.
向量是 GCSE OCR 数学大纲的基础内容，同时出现在基础和高级试卷中。理解向量对于解决几何问题、证明共线性以及处理平移变换至关重要。本指南将逐一讲解每一个核心概念，从向量的符号表示到向量证明，并配有清晰示例与应试技巧，助你全面掌握向量。
1. Vector Notation and Basics | 向量符号与基础
A vector is a quantity that has both magnitude (size) and direction; a scalar has only magnitude. In OCR exams, vectors are usually printed in bold (e.g., a) and you should underline them when handwriting. Points are labelled with capital letters, and the vector from A to B is written as AB (with an arrow above in typeset, but you can write it as vector AB). Two vectors are equal if they have the same magnitude and direction, no matter where they start.
向量是既有大小又有方向的量；标量只有大小。OCR 考试中向量通常印为粗体（如 a），手写时须在字母下方画线。点用大写字母标记，从 A 到 B 的向量记作 AB（印刷体会加箭头，你也可以写成向量 AB）。两个向量如果大小与方向都相同，便是相等向量，与起点位置无关。
For example, the vector a = 3i + 4j means a movement of 3 units right and 4 units up. In column form, it is represented vertically with 3 above 4. The zero vector 0 has magnitude zero and no specific direction. The negative of a vector, −a, has the same length as a but points in the opposite direction.
例如，向量 a = 3i + 4j 表示向右 3 个单位、向上 4 个单位的位移。列向量形式则是将 3 置于 4 上方竖直书写。零向量 0 的大小为零，没有特定方向。向量的负向量 −a 与 a 长度相同，但方向相反。
2. Column Vectors and Component Form | 列向量与分量形式
A vector can be written in column form as a 2×1 matrix. For example, a = (4, −3) in component form corresponds to a column vector with 4 at the top and −3 at the bottom, representing 4 units right and 3 units down. This is equivalent to 4i − 3j, where i is a horizontal unit vector to the right and j is a vertical unit vector upwards.
向量可以用 2×1 矩阵（列向量）表示。例如，分量形式 a = (4, −3) 对应上 4 下 −3 的列向量，表示向右 4 单位、向下 3 单位。它等价于 4i − 3j，这里 i 是向右的水平单位向量，j 是向上的竖直单位向量。
You can switch between the two notations freely: the column vector with entries x and y is exactly xi + yj. Use whichever the question demands, but i–j form is often easier for algebraic manipulation.
你可以在两种表示之间自由切换：分量为 x 和 y 的列向量就是 xi + yj。按照题目要求选用，但 i–j 形式在代数运算中往往更方便。
3. Vector Addition and Subtraction | 向量的加法与减法
To add two vectors, simply add their corresponding components. If a = a₁i + a₂j and b = b₁i + b₂j, then a + b = (a₁ + b₁)i + (a₂ + b₂)j. Geometrically, this follows the triangle law: place the tail of b at the head of a; the resultant vector goes from the tail of a to the head of b.
两个向量相加，只需将对应分量相加。若 a = a₁i + a₂j，b = b₁i + b₂j，则 a + b = (a₁ + b₁)i + (a₂ + b₂)j。几何上遵循三角形法则：把 b 的尾端置于 a 的尖端，合向量便从 a 的尾指向 b 的头。
Subtraction a − b is defined as a + (−b), so subtract the components: (a₁ − b₁)i + (a₂ − b₂)j. The parallelogram law is an alternative geometric view: the diagonals of a parallelogram represent a + b and a − b.
向量减法 a − b 定义为 a + (−b)，因此对应分量相减：(a₁ − b₁)i + (a₂ − b₂)j。平行四边形法则是另一种几何视角：平行四边形的对角线分别表示 a + b 与 a − b。
4. Scalar Multiplication | 标量乘法
Multiplying a vector by a scalar λ stretches or compresses its length and may reverse its direction. If λ > 0, the direction stays the same; if λ < 0, the direction is reversed. The magnitude is scaled by |λ|. In components: λa = (λa₁)i + (λa₂)j.
标量 λ 乘以一个向量会拉伸或压缩其长度，并可能反转方向。λ > 0 时方向不变；λ < 0 时方向反转。大小按 |λ| 缩放。分量形式为 λa = (λa₁)i + (λa₂)j。
Two vectors are parallel if one is a scalar multiple of the other: b = ka for some non‑zero scalar k. This fact is essential for proving collinearity and parallel lines.
如果两个向量中一个是另一个的标量倍，即 b = ka（k 为非零标量），则它们平行。这一事实是证明共线和平行的关键。
5. Magnitude of a Vector | 向量的模长
The magnitude (or modulus) of a vector a = xi + yj is found by Pythagoras’ theorem: |a| = √(x² + y²). For instance, the magnitude of 3i + 4j is √(3² + 4²) = √25 = 5. The magnitude is always a non‑negative scalar.
向量 a = xi + yj 的模长（大小）由勾股定理求得：|a| = √(x² + y²)。例如，3i + 4j 的模长为 √(3² + 4²) = √25 = 5。模长总是一个非负的标量。
A unit vector has magnitude 1. To obtain a unit vector in the direction of a, divide a by its magnitude: â = a / |a|. Unit vectors are used to specify direction without changing scale.
单位向量的模为 1。要得到与 a 同方向的单位向量，用 a 除以其模长：â = a / |a|。单位向量常用于仅表示方向而不改变尺度。
6. Position Vectors | 位置向量
The position vector of a point A is the vector from the origin O to A, written as OA or simply a. If A has coordinates (x, y), its position vector is a = xi + yj. This is the most convenient way to link coordinates and vectors.
点 A 的位置向量是指从原点 O 到 A 的向量，记作 OA 或简写为 a。若 A 的坐标为 (x, y)，其位置向量为 a = xi + yj。这是连接坐标与向量最便捷的方式。
The vector from point A to point B is AB = b − a, where a and b are the position vectors of A and B. This rule is the foundation for almost every vector path problem.
从 A 到 B 的向量是 AB = b − Published by TutorHao | GCSE Mathematics Revision Series | aleveler.com
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