📚 G-7 Mathematics: Vectors and Transformations | G-7 数学：向量与变换
In this revision article, we explore vectors and transformations, two fundamental topics in IGCSE Mathematics. You will learn the key definitions, operations, and methods needed to solve exam-style problems with confidence.
本篇复习文章将深入探讨向量与变换，这是IGCSE数学中的两个基础主题。你将学到关键的定义、运算方法，以及如何从容解答考试型题目。
1. Introduction to Vectors | 向量基础
A vector is a quantity that has both magnitude and direction. It is represented by a directed line segment, often written as a column vector such as (3, 4), where 3 is the horizontal component and 4 is the vertical component.
向量是既有大小又有方向的量。它通常用有向线段表示，也常写为列向量，例如 (3, 4)，其中3是水平分量，4是垂直分量。
Two vectors are equal if they have the same length and direction, regardless of their starting point. A vector can also be named using a bold letter like v, or by a pair of coordinates.
若两个向量具有相同的长度和方向，则它们相等，与起点无关。向量也可用粗体字母如 v 或一对坐标表示。
2. Vector Addition and Subtraction | 向量加减法
To add vectors, we add their corresponding components. For example, (a, b) + (c, d) = (a + c, b + d). This is equivalent to placing the tail of one vector at the head of the other.
向量加法只需将对应分量相加。例如，(a, b) + (c, d) = (a + c, b + d)。这等价于把第二个向量的起点放在第一个向量的终点。
Subtraction is performed similarly: (a, b) − (c, d) = (a − c, b − d). Geometrically, subtracting vectors gives the vector from the head of the second to the head of the first.
向量减法同理：(a, b) − (c, d) = (a − c, b − d)。在几何上，相减得到从第二个向量终点指向第一个向量终点的向量。
3. Scalar Multiplication | 标量乘法
Multiplying a vector by a scalar changes its magnitude but keeps its direction (or reverses it if the scalar is negative). For a scalar k, k(a, b) = (ka, kb).
向量乘以标量会改变其大小，但方向不变；若标量为负数，则方向反转。对于标量 k，k(a, b) = (ka, kb)。
For example, 3(2, −1) = (6, −3). If k = −1, the vector is reflected through the origin, giving the opposite vector.
例如，3(2, −1) = (6, −3)。当 k = −1 时，向量关于原点反射，得到相反向量。
4. Position Vectors | 位置向量
A position vector is a vector that starts from the origin O and ends at a given point P. If P has coordinates (x, y), its position vector is OP = (x, y).
位置向量是从原点 O 出发，指向给定点 P 的向量。若 P 的坐标为 (x, y)，则其位置向量为 OP = (x, y)。
Position vectors are often used to find the vector between two points. The vector from A to B is OB − OA = (x₂ − x₁, y₂ − y₁).
位置向量常用于求两点之间的向量。从 A 到 B 的向量为 OB − OA = (x₂ − x₁, y₂ − y₁)。
5. Magnitude of a Vector | 向量的模
The magnitude or length of a vector (x, y) is found using Pythagoras’ theorem. The formula is:
向量 (x, y) 的模或长度通过勾股定理求出，公式如下：
|v| = √(x² + y²)
For example, the vector (3, 4) has magnitude √(9 + 16) = √25 = 5. A unit vector is a vector of magnitude 1, used to indicate direction only.
例如，向量 (3, 4) 的模为 √(9 + 16) = √25 = 5。单位向量是模为1的向量，仅表示方向。
6. Translations | 平移
A translation moves every point of a shape by the same vector. If a point (x, y) is translated by vector (a, b), the image is (x + a, y + b).
平移是使图形上每一个点都沿同一向量移动。若点 (x, y) 按向量 (a, b) 平移，其像点为 (x + a, y + b)。
In exam questions, you may be given a shape and a translation vector, and asked to draw the image. Remember that translation does not change the size, shape, or orientation of the object.
考试中可能会给出图形和平移向量，要求画出像。注意平移不改变物体的大小、形状或方向。
7. Reflections | 反射
A reflection mirrors a shape across a line, called the line of reflection. Common lines include the x-axis, y-axis, y = x, and x = c.
反射是将图形关于一条直线作镜像，该直线称为反射轴。常见的反射轴有 x 轴、y 轴、y = x 和 x = c。
For reflection in the x-axis, the y-coordinate changes sign: (x, y) → (x, −y). In the y-axis: (x, y) → (−x, y). In y = x: (x, y) → (y, x).
关于 x 轴对称时，纵坐标变号：(x, y) → (x, −y)。关于 y 轴对称：(x, y) → (−x, y)。关于 y = x 对称：(x, y) → (y, x)。
8. Rotations | 旋转
A rotation turns a shape about a fixed point called the centre of rotation. The angle of rotation is often 90°, 180°, or 270°, and may be clockwise or anticlockwise.
旋转是图形绕固定点（旋转中心）转动。旋转角通常为 90°、180° 或 270°，方向可为顺时针或逆时针。
To rotate a point (x, y) 90° clockwise about the origin, the image is (y, −x). For 180° about the origin: (x, y) → (−x, −y). For 90° anticlockwise: (x, y) → (−y, x).
将点 (x, y) 绕原点顺时针旋转 90°，像为 (y, −x)。绕原点 180°： (x, y) → (−x, −y)。逆时针 90°： (x, y) → (−y, x)。
9. Enlargements | 放大
An enlargement changes the size of a shape by a scale factor k, from a fixed centre of enlargement. If k > 1, the shape becomes larger; if 0 < k < 1, it becomes smaller. A negative scale factor also rotates the shape by 180°.
放大是按比例系数 k 以固定中心改变图形大小。当 k > 1 时图形变大；当 0 < k < 1 时图形变小。负比例系数还会使图形旋转 180°。
The distance from the centre to each vertex is multiplied by k. The area of the image is k² times the original area, while side lengths are k times the original.
从中心到每个顶点的距离乘以 k。像的面积是原面积的 k² 倍，而边长是原来的 k 倍。
10. Combined Transformations and Matrices | 复合变换与矩阵
Two or more transformations can be applied in succession. The order matters. For example, reflecting first and then rotating is generally different from rotating first and then reflecting.
多个变换可以依次进行。顺序至关重要。例如，先反射再旋转通常不同于先旋转再反射。
Transformations can be represented by 2×2 matrices. Reflection in the x-axis uses matrix:
变换可以用 2×2 矩阵表示。关于 x 轴的反射矩阵为：
| 1 | 0 |
| 0 | −1 |
Rotation 90° anticlockwise uses matrix:
逆时针旋转 90° 的矩阵为：
| 0 | −1 |
| 1 | 0 |
To combine transformations, multiply the matrices in reverse order. If a rotation R is followed by a reflection S, the combined matrix is S × R.
复合变换时，按相反顺序将矩阵相乘。若先旋转 R 后反射 S，则复合矩阵为 S × R。
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