IGCSE Maths Topic 7: Vectors and Transformations | IGCSE 数学第7章：向量与变换

📚 IGCSE Maths Topic 7: Vectors and Transformations | IGCSE 数学第7章：向量与变换

Vectors and transformations are central to the IGCSE Mathematics syllabus because they connect algebra with geometry and coordinate methods. This guide is designed for teachers and students who want a clear, exam-focused review of Topic 7, including definitions, techniques, and common errors.

向量与变换是 IGCSE 数学大纲的核心内容，因为它们将代数、几何与坐标方法紧密联系起来。本指南面向教师和学生，提供第7章的高效复习，包括定义、解题方法以及常见错误。

1. Understanding Vectors | 理解向量

A vector is a quantity that has both magnitude and direction. Examples include displacement, velocity, and force. A scalar has magnitude only, such as distance, speed, and mass. In IGCSE questions, you must be able to decide whether a given quantity is a vector or a scalar.

向量是既有大小又有方向的量，例如位移、速度和力。标量只有大小，例如路程、速率和质量。在 IGCSE 考试中，你必须能够判断给定的量是向量还是标量。

Vectors are not tied to a fixed position unless they are position vectors. A free vector can be moved parallel to itself without changing its meaning. This idea is essential when adding vectors or proving that two lines are parallel.

向量通常不固定在某个位置，除非它是位置向量。自由向量可以平行移动而不改变其含义。这个思想在向量加法或证明两条线平行时非常重要。

2. Vector Notation and Representation | 向量表示法

In IGCSE, a vector is often written as a bold letter, such as a, or as a column vector. For example, the vector (3, 4) means 3 units to the right and 4 units upwards. The top number gives the horizontal component, and the bottom number gives the vertical component.

在 IGCSE 中，向量通常用粗体字母表示，例如 a，或者用列向量表示。例如，向量 (3, 4) 表示向右 3 个单位、向上 4 个单位。上方的数字表示水平分量，下方的数字表示垂直分量。

A vector between two points A and B can be written as AB. The direction matters: AB goes from A to B, while BA has the same length but the opposite direction, so BA = −AB.

点 A 到点 B 之间的向量可以写作 AB。方向很重要：AB 从 A 指向 B，而 BA 长度相同但方向相反，因此 BA = −AB。

3. Adding and Subtracting Vectors | 向量加减法

To add two vectors, add their corresponding components. If a = (2, 3) and b = (−1, 5), then a + b = (2 + (−1), 3 + 5) = (1, 8). Geometrically, vector addition follows the triangle law: place the tail of the second vector at the head of the first, then draw the resultant from the start to the final point.

两个向量相加时，只需将对应的分量相加。如果 a = (2, 3) 且 b = (−1, 5)，则 a + b = (2 + (−1), 3 + 5) = (1, 8)。从几何上看，向量加法遵循三角形法则：将第二个向量的起点放在第一个向量的终点，然后从起点到最终点画出合向量。

Subtraction is equivalent to adding the negative vector. For example, a − b = a + (−b). If b = (−1, 5), then −b = (1, −5), so a − b = (2 + 1, 3 + (−5)) = (3, −2). Always change the sign of the vector being subtracted.

向量减法等价于加上负向量。例如，a − b = a + (−b)。如果 b = (−1, 5)，那么 −b = (1, −5)，所以 a − b = (2 + 1, 3 + (−5)) = (3, −2)。被减向量的符号要全部改变。

4. Scalar Multiplication of Vectors | 向量数乘

Scalar multiplication means multiplying a vector by a number. To find kv, multiply each component of v by k. For example, if v = (2, −3), then 3v = (6, −9). The vector remains parallel to the original, but its magnitude changes by a factor of |k|.

数乘向量是指用一个数乘以一个向量。计算 kv 时，将 v 的每个分量都乘以 k。例如，若 v = (2, −3)，则 3v = (6, −9)。向量方向仍与原向量平行，但大小变为原来的 |k| 倍。

If k is negative, the direction is reversed. For example, −2v = (−4, 6) when v = (2, −3). This is useful when simplifying vector expressions such as 2a + 3b − a = a + 3b.

如果 k 为负数，方向会反转。例如，当 v = (2, −3) 时，−2v = (−4, 6)。这在化简诸如 2a + 3b − a = a + 3b 的向量表达式时非常有用。

5. Magnitude of a Vector | 向量的模

The magnitude of a vector v = (x, y) is its length, calculated using Pythagoras' theorem: |v| = √x² + y². For example, the magnitude of (3, 4) is √3² + 4² = √25 = 5.

向量 v = (x, y) 的模是其长度，用勾股定理计算：|v| = √x² + y²。例如，向量 (3, 4) 的模为 √3² + 4² = √25 = 5。

Magnitude is always non-negative. It is used in problems involving distances, speeds, and unit vectors. A unit vector has magnitude 1 and is found by dividing a vector by its magnitude.

模总是非负的。它常用于距离、速率和单位向量的相关题目。单位向量的模为 1，可以用向量除以它的模得到。

6. Position Vectors | 位置向量

A position vector starts at the origin O and ends at a point P. If P has coordinates (x, y), then the position vector OP = (x, y). Position vectors allow us to describe points using vector notation.

位置向量从原点 O 出发，终点为点 P。如果 P 的坐标为 (x, y)，则位置向量 OP = (x, y)。位置向量使我们能够用向量记号表示点。

To find the vector AB from the position vectors of A and B, use the rule AB = OB − OA. For example, if OA = (1, 2) and OB = (5, 4), then AB = (5 − 1, 4 − 2) = (4, 2).

要从 A 点和 B 点的位置向量求向量 AB，使用公式 AB = OB − OA。例如，若 OA = (1, 2) 且 OB = (5, 4)，则 AB = (5 − 1, 4 − 2) = (4, 2)。

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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, u = (3, 6) and v = (1, 2) are parallel because u = 3v. When vectors are written in column form, check that the ratios of corresponding components are equal.

如果一个向量是另一个向量的数乘，则这两个向量平行。例如，u = (3, 6) 与 v = (1, 2) 平行，因为 u = 3v。当向量写成列向量形式时，可检查对应分量的比值是否相等。

To prove three points A, B, and C are collinear, show that AB and BC are parallel and share a common point B. For instance, if AB = (2, 4) and BC = (1, 2), then AB = 2BC, so A, B, C lie on the same straight line.

要证明三点 A、B、C 共线，只需证明 AB 与 BC 平行且有一个公共点 B。例如，若 AB = (2, 4) 且 BC = (1, 2)，则 AB = 2BC，因此 A、B、C 在同一直线上。

8. Introduction to Transformations | 变换入门

A transformation maps an object to an image on the coordinate plane. The four main IGCSE transformations are translation, reflection, rotation, and enlargement. Some transformations preserve length and angle; these are called isometries.

变换将坐标平面上的原图形映射为像图形。IGCSE 中四种主要变换是平移、反射、旋转和放大。有些变换保持长度和角度不变，这类变换称为等距变换。

Translation, reflection, and rotation are isometries, so the object and image are congruent. Enlargement changes size but preserves shape, so the object and image are similar rather than congruent.

平移、反射和旋转都是等距变换，因此原图形与像图形全等。放大会改变大小但保持形状，所以原图形与像图形相似但不全等。

9. Translations | 平移

A translation moves every point of a shape by the same vector. It is described by a translation vector such as (4, −2), meaning 4 units right and 2 units down. The translated image has the same orientation as the object.

平移将图形的每个点按同一个向量移动。它用平移向量描述，例如 (4, −2) 表示向右 4 个单位、向下 2 个单位。平移后的像图形方向与原图形相同。

To perform a translation on a point, add the translation vector to the position vector of the point. For example, point A (1, 3) translated by (4, −2) becomes A′ (5, 1). Translation never changes the shape or size.

对点进行平移时，将平移向量与该点的位置向量相加。例如，点 A (1, 3) 按 (4, −2) 平移后变为 A′ (5, 1)。平移不会改变形状或大小。

10. Reflections | 反射

same perpendicular distance from the mirror line as the object, but on the opposite side.

反射将图形沿对称轴翻转。IGCSE 中常见的对称轴有 x 轴、y 轴、直线 y = x 和直线 y = −x。像图形到对称轴的垂直距离与原图形相同，但位于对称轴的另一侧。

Mirror line | 对称轴	Mapping rule | 映射规则
x-axis	(x, y) → (x, −y)
y-axis	(x, y) → (−x, y)
y = x	(x, y) → (y, x)
y = −x	(x, y) → (−y, −x)

Reflections produce congruent images, but orientation is reversed. When describing a reflection, always state the equation of the mirror line, not just "across the middle".

反射产生全等的像，但方向相反。描述反射时，一定要写出对称轴的方程，而不能只说“沿中间翻转”。

11. Rotations | 旋转

A rotation turns a shape about a fixed centre by a specified angle and direction. To describe a rotation fully, give the centre, angle, and direction, for example "rotation of 90° clockwise about the origin".

旋转是围绕固定中心按指定角度和方向转动图形。要完整描述旋转，需给出旋转中心、角度和方向，例如“绕原点顺时针旋转 90°”。

nd 180°: (x, y) → (−x, −y). Rotation preserves distance, so it produces a congruent image.

IGCSE 中常见的绕原点旋转包括：顺时针 90°：(x, y) → (y, −x)；逆时针 90°：(x, y) → (−y, x)；180°：(x, y) → (−x, −y)。旋转保持距离不变，因此产生全等的像。

12. Enlargements | 放大

An enlargement changes the size of a shape by a scale factor k relative to a centre of enlargement. If k > 1, the image is larger; if 0 < k < 1, the image is smaller. A negative scale factor also rotates the image by 180°, but this is usually examined at extended level.

放大是以放大中心为基准，按比例因子 k 改变图形的大小。如果 k > 1，像图形变大；如果 0 < k < 1，像图形变小。负比例因子还会将像图形旋转 180°，但这通常在扩展级别考查。

kx, ky). For enlargement from a non-origin centre, first subtract the centre, multiply by k, then add the centre back. Enlargement produces a similar image with angles unchanged.

从原点放大一个点时，将其坐标乘以 k：(x, y) → (kx, ky)。若放大中心不是原点，则先减去中心坐标，再乘以 k，最后加回中心坐标。放大产生相似形，角度不变。

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