GCSE AQA Maths: Vectors Complete Revision Guide | GCSE AQA 数学：向量考点精讲

📚 GCSE AQA Maths: Vectors Complete Revision Guide | GCSE AQA 数学：向量考点精讲

Vectors are a key topic in the GCSE AQA Mathematics specification. They allow us to describe both magnitude and direction, unlike scalar quantities. This guide covers every essential concept, from basic notation to geometric proofs, ensuring you can tackle any exam question with confidence.

向量是 GCSE AQA 数学大纲中的核心考点。它同时描述大小和方向，与标量不同。本指南涵盖从基本表示法到几何证明的所有重要概念，助你自信应对任何考试题型。

1. What is a Vector? | 什么是向量？

A vector is a quantity that has both size (magnitude) and direction. In contrast, a scalar only has magnitude, such as speed, distance or mass. In GCSE Maths, vectors are used to describe translations, movements between points, and to solve geometric problems without needing coordinates.

向量是既有大小（模）又有方向的量。标量只有大小，如速率、距离或质量。在 GCSE 数学中，向量用于描述平移、点之间的移动，并可在不需要坐标的情况下解决几何问题。

2. Vector Notation | 向量表示法

Vectors can be written in several ways. A vector from point A to point B is written as AB→. In printed text, a single letter vector is often shown in bold, like a. For handwritten work, you should underline the letter: a. Column vectors show the horizontal component on top and the vertical component below, for example a vector 3 right and 4 down would be written as (3-4). AQA expects you to use and interpret both column vectors and component notation.

向量有几种写法。从点 A 到点 B 的向量记作 AB→。印刷体中单字母向量通常用粗体表示，如 a。手写时应在字母下方加下划线：a。列向量将水平分量写在上方，垂直分量写在下方，例如向右 3、向下 4 的向量写作 (3-4)。AQA 要求你会使用并理解列向量和分量表示法。

3. Scalar Multiplication | 标量乘法

Multiplying a vector by a scalar (a number) changes its magnitude but keeps its direction unchanged, unless the scalar is negative – then the direction reverses. For a column vector v = (xy), the scalar product kv = (kxky). For instance, if v = (23) and k = 3, then 3v = (69). This is essential for identifying parallel vectors.

向量乘以标量（一个数）会改变模长，但方向不变，除非标量为负数——此时方向反转。对于列向量 v = (xy)，标量积 kv = (kxky)。例如，若 v = (23)，k = 3，则 3v = (69)。这是识别平行向量的基础。

4. Vector Addition | 向量加法

To add two vectors, simply add their corresponding components. For column vectors: (ab) + (cd) = (a+cb+d). Geometrically, this follows the ‘nose-to-tail’ rule: place the tail of the second vector at the head of the first. The resultant vector goes from the tail of the first to the head of the second. Vector addition is commutative, so a + b = b + a.

向量相加，只需将对应分量相加。列向量形式：(ab) + (cd) = (a+cb+d)。几何上遵循“首尾相接”法则：将第二个向量的起点放在第一个向量的终点处。合向量从第一个向量的起点指向第二个向量的终点。向量加法满足交换律，即 a + b = b + a。

5. Vector Subtraction | 向量减法

Subtracting a vector is the same as adding its negative. a − b = a + (−b). In component terms: (ab) − (cd) = (a−cb−d). If two vectors start at the same origin, then a − b is the vector from the tip of b to the tip of a. This is especially helpful when finding missing sides in triangle or parallelogram questions.

向量减法等同于加上该向量的负向量。a − b = a + (−b)。分量形式：(ab) − (cd) = (a−cb−d)。若两向量起点相同，则 a − b 是从 b 的终点指向 a 终点的向量。这在求解三角形或平行四边形中缺失边时尤为有用。

6. Position Vectors | 位置向量

A position vector tells you the location of a point relative to the origin O. If point A has coordinates (x, y), its position vector is OA→ = (xy). Any vector between two points can be expressed using position vectors: AB→ = OB→ − OA→ = b − a, where a and b are the position vectors of A and B. This formula is central to many GCSE vector proof questions.

位置向量表示点相对于原点 O 的位置。若点 A 坐标为 (x, y)，其位置向量为 OA→ = (xy)。任意两点间的向量都可以用位置向量表示：AB→ = OB→ − OA→ = b − a，其中 a、b 分别是 A、B 的位置向量。该公式是许多 GCSE 向量证明题的核心。

7. Parallel Vectors and Collinearity | 平行向量与共线

Two vectors are parallel if one is a scalar multiple of the other. For example, if u = 2v, then u and v are parallel and have the same direction. If the scalar is negative, they point in opposite directions but are still parallel. To prove that three points A, B and C are collinear (lie on a straight line), you can show that AB→ is parallel to BC→, i.e. there exists a scalar k such that AB→ = kBC→. Don’t forget to check that the point B is common to both vectors.

若一个向量是另一个向量的标量倍数，则二者平行。例如，若 u = 2v，则 u 与 v 平行且同向。若标量为负，则方向相反，但仍平行。要证明三点 A、B、C 共线，可证明 AB→ 平行于 BC→，即存在标量 k 使得 AB→ = kBC→。不要忘记检查点 B 为两向量共用点。

8. Magnitude of a Vector | 向量的模

The magnitude (length) of a vector v = (x更多咨询请联系16621398022（同微信）