📚 CIE GCSE Maths: Vectors – Essential Revision | CIE GCSE 数学：向量考点精讲
Vectors are an essential topic in the CIE GCSE Mathematics syllabus, appearing regularly in both Core and Extended level papers. They provide a powerful way to describe magnitude and direction, and are used to solve geometric problems involving lines, triangles, and parallelograms. Mastering vector notation, operations, and geometric applications is crucial for achieving a top grade. This article breaks down every key concept, offers exam tips, and includes numerous worked examples to help you revise effectively.
向量是 CIE GCSE 数学大纲中的核心考点，在核心卷和 extended 卷中均频繁出现。向量能够同时描述大小与方向，是解决直线、三角形与平行四边形等几何问题的重要工具。熟练掌握向量的表示法、运算法则以及几何应用，是取得高分的关键。本文逐一解析每个重要知识点，提供应考策略与大量范例，帮助你高效复习。
1. Vector Notation and Representation | 向量表示法
In CIE exams, vectors are typically written in bold type, such as a, or with an arrow above, a⃗. A vector can be described by its components: a column vector like (3, 4) represents 3 units to the right and 4 units up. The same vector can be expressed using unit vectors: 3i + 4j, where i = (1, 0) and j = (0, 1). Always give your answers in the form required by the question, but be comfortable converting between column vectors and i–j notation.
在 CIE 考试中，向量通常以粗体字母 a 或带箭头的 a⃗ 表示。向量可以由分向量描述：列向量 (3, 4) 表示水平方向 3 个单位，竖直方向 4 个单位。同一向量也可用单位向量表示为 3i + 4j，其中 i = (1, 0) 为水平单位向量，j = (0, 1) 为竖直单位向量。答题时应按照题目要求的形式给出答案，同时要熟练在列向量与 i–j 表示法之间切换。
| Column vector: v = (x, y) | 列向量: v = (x, y) |
| Unit vector form: v = xi + yj | 单位向量式: v = xi + yj |
| A vector has both magnitude and direction; a scalar has only magnitude. | 向量既有大小又有方向；标量只有大小。 |
2. Magnitude of a Vector | 向量的模
The magnitude (or length) of a vector a = (x, y) is given by |a| = √(x² + y²). This follows directly from Pythagoras’ theorem. For example, if a = (3, 4), then |a| = √(3² + 4²) = 5. In the i–j notation, the magnitude of a = xi + yj is still |a| = √(x² + y²). You must always give the magnitude as a positive value or zero. If the vector is given as a directed line segment AB, its length is the distance between A and B, which is exactly the magnitude of the vector AB.
向量 a = (x, y) 的模（即长度）由公式 |a| = √(x² + y²) 给出，这直接源于勾股定理。例如，若 a = (3, 4)，则 |a| = √(3² + 4²) = 5。在单位向量表示法中，a = xi + yj 的模仍为 √(x² + y²)。模必须取非负值。若以有向线段形式给出向量 AB，其长度就是 A、B 两点间的距离，也正是向量 AB 的模。
|a| = √(x² + y²)
| a = (5, 12) → |a| = √(25 + 144) = 13 | a = (5, 12) → |a| = √(25 + 144) = 13 |
3. Scalar Multiplication | 标量乘法
Multiplying a vector by a scalar (a real number) changes its length but not its direction, unless the scalar is negative, in which case the direction is reversed. Algebraically, if a = (x, y) and k is a scalar, then ka = (k x, k y). For instance, 3(1, −2) = (3, −6). In geometry, this is used to express parallel vectors: two vectors a and b are parallel if a = kb for some non‑zero scalar k. The value of k gives the ratio of their lengths.
向量乘以一个标量（实数）会改变其长度，但方向不变（除非标量为负，则方向反向）。代数上，若 a = (x, y)，k 为标量，则 ka = (k x, k y)。例如 3(1, −2) = (3, −6)。在几何中，这一性质用于表示平行向量：若存在非零实数 k 使得 a = kb，则两向量平行。k 的值给出了长度比。
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