📚 IGCSE Mathematics: Vectors and Geometric Transformations | IGCSE数学：平面向量与几何变换
Vectors and geometric transformations form a unified visual language in mathematics. A vector carries both magnitude and direction, while a transformation moves or reshapes a figure according to precise rules. Mastering these tools allows you to describe motion, position, and symmetry elegantly and quantitatively.
向量与几何变换在数学中构成一套统一的视觉语言。向量同时具有大小和方向，而变换则按精确规则移动或重塑图形。掌握这些工具，你能以优雅而量化的方式描述运动、位置和对称性。
1. What Are Vectors | 什么是向量
A vector is a quantity that has both magnitude (size) and direction. Displacement, velocity, and force are classic physical examples. In contrast, a scalar has only magnitude, such as temperature or speed.
向量是同时具有大小和方向的量。位移、速度和力都是典型的物理实例。相比之下，标量只有大小，例如温度或速率。
Two vectors are equal if they have the same magnitude and the same direction, even if they start from different points. This property makes vectors free to move in a plane, which is the basis of vector geometry.
两个向量如果大小相等且方向相同，则它们相等，即使起点不同也是如此。这一性质使向量可以在平面内自由平移，这正是向量几何的基础。
2. Representing Vectors | 向量的表示
In printed text, a vector can be written as a bold lowercase letter such as a or b. In handwriting, you underline the letter or draw an arrow above it. The vector from point A to point B is written as \overrightarrow{AB} in notation — but here we denote it using the Unicode arrow: A⃗B.
在印刷文本中，向量可写为粗体小写字母，如 a 或 b。手写时通常在字母下方加横线或在字母上方画箭头。从点A到点B的向量在记法中写为A⃗B。
Algebraically, a two-dimensional vector is shown as a column of two numbers: the top number is the horizontal component, and the bottom number is the vertical component. For example, the vector from A(1, 2) to B(4, 6) is written as:
在代数形式上，二维向量写作两行一列：上方数字为水平分量，下方数字为垂直分量。例如从A(1, 2)到B(4, 6)的向量写为：
B − A = ( 4−1 , 6−2 ) = ( 3 , 4 )
In column form this is (3 over 4), meaning move 3 units right and 4 units up.
列向量形式为（上3下4），表示向右移动3个单位，再向上移动4个单位。
3. Vector Addition and Subtraction | 向量加法与减法
To add two vectors, add their corresponding components. For example, if a = (2, 3) and b = (1, 4), then a + b = (2+1, 3+4) = (3, 7).
两个向量相加时，对应分量分别相加。例如若 a = (2, 3)，b = (1, 4)，则 a + b = (2+1, 3+4) = (3, 7)。
Geometrically, place the tail of b at the head of a. The sum vector runs from the tail of a to the head of b. This is called the triangle law of addition.
几何上，将 b 的起点置于 a 的终点，和向量从 a 的起点指向 b 的终点，这称为向量加法的三角形法则。
To subtract b from a, add the negative of b. The negative of a vector has the same magnitude but opposite direction. Component-wise, a − b = (2−1, 3−4) = (1, −1).
从 a 中减去 b，相当于加上 b 的负向量。负向量与原向量大小相等、方向相反。按分量计算，a − b = (2−1, 3−4) = (1, −1)。
4. Scalar Multiplication | 标量倍数
Multiplying a vector by a scalar scales its magnitude but leaves its direction unchanged (unless the scalar is negative, which reverses direction). If a = (2, −1), then 3a = (6, −3) and −2a = (−4, 2).
向量乘以标量会改变其大小，但方向不变（除非标量为负，此时方向反转）。若 a = (2, −1)，则 3a = (6, −3)，−2a = (−4, 2)。
Two vectors are parallel if one is a scalar multiple of the other. This is one of the most frequently tested ideas in IGCSE vector problems.
若一个向量是另一个向量的标量倍数，则这两个向量平行。这是IGCSE向量题中最常考的思想之一。
5. Position Vectors | 位置向量
The position vector of a point P is the vector from the origin O to P. If P has coordinates (x, y), then its position vector p = (x, y). Position vectors fix a point in space relative to a reference frame.
点P的位置向量是从原点O指向P的向量。若P的坐标为(x, y)，则其位置向量 p = (x, y)。位置向量将点相对于参考系固定在空间中。
For two points A and B with position vectors a and b, the displacement from A to B is b − a. This simple formula is powerful: it converts coordinates into a directed displacement.
对于位置向量分别为 a 和 b 的两点A和B，从A到B的位移为 b − a。这个简单公式很有威力：它把坐标转化为有向位移。
The midpoint M of AB has position vector (a + b)/2, found by averaging the position vectors of A and B.
AB的中点M的位置向量为 (a + b)/2，即取A和B位置向量的平均值。
6. Magnitude of a Vector | 向量的模长
The magnitude (or modulus) of a vector v = (x, y) is its length, calculated using Pythagoras’ theorem:
向量 v = (x, y) 的模长（或长度）通过勾股定理计算：
|v| = √(x² + y²)
For example, the magnitude of (3, 4) is √(9 + 16) = √25 = 5. This is the Euclidean distance from the origin to the point (3, 4).
例如 (3, 4) 的模长为 √(9 + 16) = √25 = 5，这是原点(0,0)到点(3, 4)的欧几里得距离。
Magnitude is a scalar. It tells you how far the vector carries you, regardless of direction. Unit vectors have magnitude exactly equal to 1.
模长是标量，告诉你向量携带你运动多远，与方向无关。单位向量的模长恰好为1。
7. Vectors in Geometry Problems | 向量在几何问题中的应用
Vectors solve geometry problems involving ratio and parallel lines. Suppose point P divides the line segment AB in the ratio m : n. Then the position vector of P is:
向量可解决涉及比例和平行线的几何问题。设点P按比例 m : n 分割线段AB，则P的位置向量为：
p = (n×a + m×b) / (m + n)
For example, if m : n = 1 : 2, then P is one-third of the way from A to B, and p = (a + 2b)/3. Check: when m = 1, P is closer to A.
例如，若 m : n = 1 : 2，则P位于从A到B的三分之一处，p = (a + 2b)/3。检验：当 m = 1 时，P更靠近A。
If three points are collinear, the vectors connecting them are scalar multiples of each other. This criterion is used to prove collinearity and parallelism in exam questions.
若三点共线，则连接它们的向量互为标量倍数。此准则常用于证明考试题中的共线与平行关系。
8. Geometric Transformations | 几何变换概述
A geometric transformation maps every point of a figure to a new position. The image is the result of the transformation, and the original figure is called the object. Four basic isometries (transformations preserving distance) are translation, reflection, rotation, and enlargement.
几何变换将图形的每一点映射到新位置。变换后的图形称为像，原图形称为原像。四种基本的等距变换（保持距离的变换）是平移、反射、旋转和放大（缩放）。
Each transformation has a precise definition. In IGCSE examinations, you must describe a transformation completely: direction, distance, line, angle, centre, or scale factor. Missing one detail loses marks.
每种变换都有精确定义。在IGCSE考试中，你必须完整描述变换：方向、距离、直线、角度、中心或比例因子。漏掉一个细节就会失分。
9. Translation | 平移变换
A translation slides every point of the object by the same vector. The translation vector is written as a column: (x over y), where x is the horizontal movement (right positive) and y is the vertical movement (up positive).
平移将原像的每个点沿同一向量滑动。平移向量写作列向量（上x下y），其中x为水平移动量（向右为正），y为垂直移动量（向上为正）。
For example, a translation by vector (3 over −2) shifts every point 3 units right and 2 units down. The size, shape, and orientation of the figure remain unchanged.
例如，平移向量（上3下−2）将每个点向右移动3个单位，向下移动2个单位。图形的大小、形状和方向均保持不变。
To find the translation that maps A to A′, subtract the coordinates of A from A′: translation vector = A′ − A. This is identical to the displacement vector between corresponding points.
要找将A映射到A′的平移，用A′的坐标减去A的坐标：平移向量 = A′ − A。这等价于对应点之间的位移向量。
10. Reflection | 反射变换
A reflection produces a mirror image across a fixed line called the mirror line (or axis of reflection). Every point on the object is mapped to a point on the opposite side of the mirror line, at an equal perpendicular distance.
反射在称为镜面（或反射轴）的固定直线另一侧产生镜像。原像上的每个点映射到镜面另一侧、垂直距离相等的点上。
A reflection reverses orientation: a clockwise triangle becomes anticlockwise after reflection. The mirror line may be vertical (x = k), horizontal (y = k), or diagonal (y = x or y = −x).
反射会反转方向：顺时针的三角形反射后变为逆时针。镜面可以是竖直线（x = k）、水平线（y = k）或对角线（y = x 或 y = −x）。
To describe a reflection fully, you must state both the type and the equation of the mirror line. Examiners expect exact equations, not just a sketch.
要完整描述反射，你必须说明变换类型和镜面方程。阅卷者期望精确的方程，不仅仅是草图。
11. Rotation | 旋转变换
A rotation turns the object about a fixed point called the centre of rotation through a given angle. By convention, a positive angle rotates anticlockwise, and a negative angle rotates clockwise.
旋转使原像绕称为旋转中心的固定点转动给定角度。按惯例，正角度为逆时针旋转，负角度为顺时针旋转。
To describe a rotation completely, you must give three details: the centre of rotation, the angle (usually 90°, 180°, or 270°), and the direction (clockwise or anticlockwise, unless the angle is 180°).
要完整描述旋转，必须给出三个要素：旋转中心、旋转角（通常为90°、180°或270°）和方向（顺时针或逆时针；若角度为180°，则方向可省略）。
For a rotation of 90° anticlockwise about the origin, the point (x, y) maps to (−y, x). For 90° clockwise about the origin, (x, y) maps to (y, −x). These shortcuts save time in examinations.
绕原点逆时针旋转90°时，点(x, y)映射为(−y, x)。绕原点顺时针旋转90°时，(x, y)映射为(y, −x)。这些快捷规则能为考试节省时间。
12. Enlargement and Combined Transformations | 放大与组合变换
An enlargement changes the size of the figure by a scale factor k, centred at a fixed point. If k > 1, the image is larger; if 0 < k < 1, smaller; if k is negative, the image is inverted and appears on the opposite side of the centre.
缩放变换以固定点为中心，按比例因子k改变图形大小。若k > 1，像变大；若0 < k < 1，像变小；若k为负，像被倒置并出现在中心的另一侧。
To find the image of point P under enlargement centre C and scale factor k, use the formula: position vector of image = C + k × (P − C). Every point moves along a straight line through the centre.
若要求点P在中心C、比例因子k的缩放下的像，使用公式：像的位置向量 = C + k × (P − C)。每个点都沿通过中心的直线移动。
When two transformations are applied in sequence, the order is critical. If an enlargement with scale factor 2 about the origin is followed by a translation (1, 3), the result is different from applying the translation first. To describe combined transformations, work step by step, tracking coordinates after each operation.
当依次进行两个变换时，次序至关重要。若绕原点比例因子为2的缩放先执行，随后平移(1, 3)，其结果与先平移再缩放完全不同。要描述组合变换，必须逐步计算，跟踪每次操作后的坐标。
A composition of transformations is not generally commutative. Exam questions often give the image coordinates and ask you to find the original object coordinates by applying the inverse of each transformation in reverse order.
变换的复合一般不满足交换律。考试题目常给出像的坐标，要求你按相反顺序求每个变换的逆变换，从而得到原像坐标。
13. Exam Tips and Common Errors | 应试要点与常见错误
First, always label vectors with bold notation in print or an arrow in handwriting. An unlabelled vector is ambiguous. Second, when subtracting vectors, write the formula b − a explicitly to avoid sign errors. Third, for transformations, always write the centre, angle, direction, or scale factor — incomplete descriptions are considered wrong.
第一，印刷体中始终用粗体标注向量，手写时加箭头。未标注的向量含义不清。第二，做向量减法时，明确写出公式 b − a，避免符号错误。第三，描述变换时，务必写明中心、角度、方向或比例因子——描述不完整将被判错。
Common errors include reversing the order of scalar multiplication, forgetting that a negative scale factor reverses orientation, and using the wrong translation vector (A′ − A instead of A − A′). Draw a sketch when in doubt — a clear diagram often reveals the correct transformation.
常见错误包括：标量乘法顺序颠倒，忘记负比例因子会反转方向，以及用错平移向量（A′ − A 与 A − A′ 混淆）。拿不准时画一画草图——清晰的图形往往能揭示正确的变换。
Finally, practise converting between column vectors and coordinate geometry. Many IGCSE questions blend these two representations, and fluency in both is the key to full marks.
最后，多练习列向量与坐标几何之间的互换。许多IGCSE题目将这两种表示融合在一起，熟悉两者是获得满分的关键。
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