📚 IGCSE Mathematics: Transformations & Vectors | 数学变换与向量
In IGCSE Mathematics, transformations and vectors form a bridge between algebra and geometry. A transformation moves or changes a shape according to a rule, while a vector is a quantity with both magnitude and direction that can describe a translation.
在IGCSE数学中，变换与向量是连接代数与几何的桥梁。变换按照某种规则移动或改变图形，而向量是同时具有大小和方向的量，可以用来描述平移。
This revision guide covers the core ideas you need for your exams: reading and writing vectors, performing reflections, rotations, translations and enlargements, combining transformations, and describing a transformation from one diagram to another.
本篇复习指南涵盖考试所需的核心知识：向量的读写、平移、反射、旋转和放大的操作、复合变换，以及如何根据图形描述一个变换。
1. Column Vectors and Magnitude | 列向量与模长
A vector describes a movement from one point to another. In two dimensions, a vector is often written as a column vector, with the top number showing the horizontal displacement and the bottom number showing the vertical displacement.
向量描述从一点到另一点的位移。在二维平面中，向量通常写成列向量的形式：上方数字表示水平位移，下方数字表示垂直位移。
column vector = ab
For example, the vector 3−2 means “3 units right and 2 units down”. A positive top number means right; a negative top number means left. A positive bottom number means up; a negative bottom number means down.
例如，向量 3−2 表示“向右3个单位，向下2个单位”。上方数字为正表示向右，为负表示向左；下方数字为正表示向上，为负表示向下。
The magnitude of a vector is its length. If the vector is xy, then its magnitude is given by Pythagoras’ theorem:
向量的模长就是它的长度。若向量为 xy，则其模长由勾股定理给出：
|v| = √(x² + y²)
For instance, the vector 34 has magnitude √(3² + 4²) = √25 = 5.
例如，向量 34 的模长为 √(3² + 4²) = √25 = 5。
Two vectors are equal if their horizontal and vertical components are both equal. The negative of a vector has the same magnitude but the opposite direction.
如果两个向量的水平分量和垂直分量分别相等，则这两个向量相等。一个向量的负向量与其模长相同，但方向相反。
2. Vector Addition, Subtraction and Scalar Multiplication | 向量的加减法与数乘
Vectors can be added by adding the corresponding components. If u = ab and v = cd, then:
向量相加时，将对应的分量相加。若 u = ab，v = cd，则：
u + v = a + cb + d
Subtraction works in the same way: u − v = a − cb − d.
减法同理：u − v = a − cb − d。
Multiplying a vector by a scalar k multiplies both components by k. This changes the length of the vector but not its direction, unless k is negative, in which case the direction is reversed.
向量乘以标量 k 时，两个分量都乘以 k。这会改变向量的长度，但不改变方向；若 k 为负数，则方向反转。
k × ab = kakb
For example, if v = 2−1, then 3v = 6−3 and −2v = −42.
例如，若 v = 2−1，则 3v = 6−3，−2v = −42。
Parallel vectors are scalar multiples of each other. This idea is often tested with vector geometry questions.
平行向量互为标量倍。这一思想经常在向量几何题中出现。
3. Translations Using Vectors | 用向量表示平移
A translation slides every point of a shape by the same vector. It does not change the size, shape or orientation of the object.
平移将图形上的每一个点沿同一个向量滑动。平移不改变图形的大小、形状或朝向。
If a point P(x, y) is translated by the vector ab, the image point P’ has coordinates:
若点 P(x, y) 沿向量 ab 平移，则像点 P’ 的坐标为：
P’ = (x + a, y + b)
Example: Translate A(2, 3) by the vector 3−2. The image is A'(2 + 3, 3 − 2) = A'(5, 1).
例题：将点 A(2, 3) 按向量 3−2 平移。像点为 A'(2 + 3, 3 − 2) = A'(5, 1)。
To find the translation vector between two corresponding points, subtract the coordinates of the object point from the image point:
要求两个对应点之间的平移向量，用像点坐标减去原图形对应点坐标：
translation vector = x′ − xy′ − y
Translation is the only transformation that uses a vector as its full description. When describing a translation, always give the vector.
平移是唯一以向量作为完整描述的变换。描述平移时，必须给出平移向量。
4. Reflections | 反射
A reflection flips a shape over a line called the mirror line. Every point and its image are at equal perpendicular distances from the mirror line.
反射是图形关于一条直线（称为镜线）的翻折。每个点到镜线的垂直距离与其像点到镜线的垂直距离相等。
Points on the mirror line do not move. They are invariant points.
位于镜线上的点在反射中位置不变，它们是“不变点”。
The table below gives the most common reflection mappings for a point (x, y).
下表给出常见的点 (x, y) 的反射变换结果。
| Mirror line | Mapping |
| x-axis | (x, y) → (x, −y) |
| y-axis | (x, y) → (−x, y) |
| y = x | (x, y) → (y, x) |
| y = −x | (x, y) → (−y, −x) |
| vertical line x = a | (x, y) → (2a − x, y) |
| horizontal line y = b | (x, y) → (x, 2b − y) |
For a reflection in a line that is not one of the axes, draw the mirror line carefully and use tracing paper or perpendicular distances.
如果镜线不是坐标轴，应仔细画出镜线，并使用透明描图纸或垂直距离来确定像点。
Reflections change orientation: the image is a mirror image, so the order of vertices written clockwise may become anticlockwise.
反射会改变图形的“朝向”：像是镜像，所以原本按顺时针排列的顶点顺序可能变为逆时针。
5. Rotations | 旋转
A rotation turns a shape about a fixed point called the centre of rotation. To describe a rotation fully, you must give the centre, the angle, and the direction (clockwise or anticlockwise).
旋转是图形绕固定点（称为旋转中心）转动。完整描述旋转需要给出旋转中心、旋转角度和旋转方向（顺时针或逆时针）。
In IGCSE, angles are often multiples of 90 degrees. The most common rotations about the origin are given below.
在IGCSE中，旋转角通常是90°的倍数。下面给出绕原点旋转的常见情况。
| Rotation | Mapping of (x, y) |
| 90° anticlockwise | (x, y) → (−y, x) |
| 90° clockwise | (x, y) → (y, −x) |
| 180° | (x, y) → (−x, −y) |
Example: Rotate the point P(4, 1) by 90° anticlockwise about the origin. The image is P'(−1, 4).
例题：将点 P(4, 1) 绕原点逆时针旋转90°。像点为 P'(−1, 4)。
For rotations about a centre other than the origin, use tracing paper, or use the rule “same distances from the centre”. You can also translate the centre to the origin, rotate, then translate back.
若旋转中心不是原点，可以使用透明描图纸，或利用“旋转中心到点及其像的距离相等”的原则。也可以先将旋转中心平移到原点，旋转后再平移回去。
The centre of rotation is invariant. No other point is invariant unless the rotation angle is 360°.
旋转中心是不变点。除非旋转角为360°，否则其他点都会移动。
6. Enlargements | 放大与缩小
An enlargement changes the size of a shape by a scale factor k, using a centre of enlargement as the fixed point.
放大（或缩小）以缩放中心为固定点，按缩放比例 k 改变图形的大小。
If the centre of enlargement is (a, b), then the image of point (x, y) is:
若缩放中心为 (a, b)，则点 (x, y) 的像为：
(x′, y′) = (a + k(x − a), b + k(y − b))
Equivalently, x′ = kx + (1 − k)a and y′ = ky + (1 − k)b.
等价地，x′ = kx + (1 − k)a，y′ = ky + (1 − k)b。
If k > 1, the shape becomes larger. If 0 < k < 1, the shape becomes smaller. If k is negative, the image is also rotated 180° about the centre.
若 k > 1，图形变大；若 0 < k < 1，图形变小；若 k 为负数，则图形还会绕缩放中心旋转180°。
The scale factor can be found by comparing corresponding lengths:
缩放比例可以通过比较对应边的长度得到：
k = image length / object length
The area scale factor is the square of the linear scale factor: area scale factor = k².
面积缩放比例是线性缩放比例的平方：面积缩放比例 = k²。
Example: A triangle with area 10 cm² is enlarged with k = 3. Its image has area 10 × 3² = 90 cm².
例题：一个面积为10 cm² 的三角形按 k = 3 放大，其像的面积为 10 × 3² = 90 cm²。
To find the centre of enlargement, draw straight lines through each point and its image. These lines intersect at the centre.
要确定缩放中心，连接每个原图形点与其对应像点，这些直线的交点就是缩放中心。
7. Composite Transformations | 复合变换
When two or more transformations are applied in order, the result is a composite transformation. The order matters: reflecting then translating can give a different result from translating then reflecting.
当按顺序应用两个或多个变换时，结果称为复合变换。变换的顺序很重要：先反射再平移，与先平移再反射，结果可能不同。
For example, let T be a translation by 20, and let R be a reflection in the y-axis. Starting with A(1, 1):
例如，设 T 为平移向量 20，R 为关于 y 轴的反射。从 A(1, 1) 开始：
-
Apply T first: A(1, 1) → A₁(3, 1). Then apply R: A₁(3, 1) → A₂(−3, 1).
先应用 T：A(1, 1) → A₁(3, 1)。再应用 R：A₁(3, 1) → A₂(−3, 1)。
-
Apply R first: A(1, 1) → A₁(−1, 1). Then apply T: A₁(−1, 1) → A₂(1, 1).
先应用 R：A(1, 1) → A₁(−1, 1)。再应用 T：A₁(−1, 1) → A₂(1, 1)。
When reading a question, follow the transformations exactly in the order given. The first transformation is applied to the original shape.
读题时，一定要严格按题目给出的顺序进行变换。第一个变换作用在原图形上。
Two successive translations can be combined into one translation by adding their vectors. Two successive enlargements with the same centre can be combined by multiplying their scale factors.
连续两次平移可以合并为一次平移，方法是把两个平移向量相加。同一中心连续两次放大可以合并为一次放大，方法是把两个缩放比例相乘。
8. Describing a Transformation | 描述一个变换
Exam questions often give an object and its image and ask you to describe the single transformation that maps one to the other. You must identify both the type of transformation and its exact details.
考试中常见题型是给出原图形和像图形，要求描述将前者变为后者的单一变换。你不仅要判断变换类型，还要写出精确的参数。
To describe a translation, find the vector that takes one vertex of the object to the corresponding vertex of the image.
描述平移时，找出从原图形一个顶点到像图形对应顶点的向量即可。
To describe a reflection, find the mirror line. Construct perpendicular bisectors between corresponding points, or look for the line that leaves the diagram symmetric.
描述反射时，需要确定镜线。可以作对应点连线的垂直平分线，或观察能使图形保持对称的直线。
To describe a rotation, find the centre of rotation, then state the angle and direction. Use tracing paper or compare the orientation of the object and image.
描述旋转时，先确定旋转中心，再说明旋转角度和方向。可以使用透明描图纸，或比较原图形与像的朝向。
To describe an enlargement, find the scale factor using corresponding lengths, then find the centre by joining corresponding points with straight lines.
描述放大（或缩小）时，先通过对应边长度求出缩放比例 k，再连接对应点找到缩放中心。
A quick check: if the orientation is the same and the size has changed, it is an enlargement. If the orientation is reversed, it is a reflection or an enlargement with a negative scale factor.
快速判断：若图形朝向不变但大小改变，则为放大或缩小；若朝向反转，则为反射或负缩放比例的放大。
9. Invariant Points | 不变点
An invariant point is a point that maps to itself under a transformation. Identifying invariant points helps describe transformations accurately.
不变点是指在变换下映射到自身的点。找出不变点有助于准确描述变换。
-
In a reflection, every point on the mirror line is invariant.
在反射中，镜线上的每一个点都是不变点。
-
In a rotation, only the centre of rotation is invariant.
在旋转中，只有旋转中心是不变点。
-
In a translation, there are no invariant points unless the translation vector is zero.
在平移中，除非平移向量为零向量，否则没有不变点。
-
In an enlargement, the centre of enlargement is invariant, but other points move unless k = 1.
在放大或缩小时，缩放中心是不变点，但其他点会移动，除非 k = 1。
Knowing invariant points can also help you determine whether a transformation is a translation, because a non-zero translation leaves no point fixed.
了解不变点还可以帮助你判断某个变换是否是平移，因为非零平移不会留下任何固定点。
10. Exam Tips and Common Mistakes | 考试技巧与常见错误
Here are the most important exam tips and common traps to avoid in transformations and vectors questions.
以下是在变换与向量题目中最需要注意的考试技巧和常见陷阱。
-
Always write column vectors in fraction-style form with the numbers directly above each other. Do not confuse the top number with the x-coordinate of a point.
始终把列向量的两个数上下对齐书写。不要把上方数字误认为点的 x 坐标。
-
When translating a shape, translate every vertex. Then join the image vertices in the same order.
平移图形时，要把每个顶点都平移，然后按相同顺序连接像顶点。
-
For reflections, check whether the image is inverted. If the mirror line is y = x, swap the coordinates.
对于反射，注意检查像是否上下或左右颠倒。若镜线为 y = x
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply