📚 Vectors: Magnitude, Addition, and Position Vectors | 向量：模长、加法与位置向量
Vectors are one of the most visual topics in IGCSE Mathematics. They allow us to describe movement, position, and direction using numbers. In this revision article, you will learn the key notation, operations, and geometric applications of vectors, with worked examples modelled on exam questions.
向量是 IGCSE 数学中最具图像化的专题之一。它用数字描述移动、位置与方向。在本复习文章中，你将学习向量的核心表示、运算与几何应用，并配有模仿真题的完整例题。
1. Scalars vs Vectors | 标量与向量
A scalar has magnitude (size) only. Examples include distance, mass, time, and temperature. A vector has both magnitude and direction. Examples include displacement, velocity, and acceleration. For instance, “5 km” is a scalar, but “5 km north” is a vector.
标量只有大小（模）。例如距离、质量、时间与温度。向量既有大小又有方向。例如位移、速度与加速度。比如“5 公里”是标量，而“5 公里向北”则是向量。
In IGCSE Mathematics, you are expected to identify whether a quantity is scalar or vector, and to use the correct language when describing motion.
在 IGCSE 数学中，你需要判断一个量是标量还是向量，并使用准确的语言描述运动。
2. Vector Notation and Column Vectors | 向量的表示与列向量
A column vector describes horizontal and vertical movement from one point to another. The vector that moves 3 units right and 2 units up is written as:
列向量用水平与垂直位移描述从一个点到另一个点的移动。向右 3 单位、向上 2 单位的向量写作：
(3, 2) or simply a = (3, 2)
The top number is the horizontal component; the bottom number is the vertical component. A vector can also be shown in bold type, for example a, or underlined. The length of a vector is shown using absolute value bars, |a|.
上方数字为水平分量，下方数字为垂直分量。向量也可用粗体（如 a）或加下划线表示。向量的模长用绝对值符号表示，即 |a|。
| Notation | Meaning | 含义 |
| (3, 2) | 3 right, 2 up | 右 3，上 2 |
| a | vector named a | 名为 a 的向量 |
| |a| | magnitude of a | 向量 a 的模长 |
3. Magnitude of a Vector | 向量的模长
The magnitude (or length) of a vector (x, y) is found using Pythagoras’ theorem:
向量 (x, y) 的模长（即长度）用勾股定理求：
|a| = √(x² + y²)
For example, for vector (3, 4): |a| = √(3² + 4²) = √(9 + 16) = √25 = 5 units. The result is always a positive scalar.
例如向量 (3, 4)：|a| = √(3² + 4²) = √(9 + 16) = √25 = 5 单位。结果始终为正标量。
This formula is essential for questions that ask for “length”, “distance”, or “magnitude”. Remember that direction is ignored when calculating magnitude.
该公式在求“长度”“距离”或“模长”时至关重要。注意计算模长时忽略方向。
4. Equality and Negative Vectors | 相等向量与负向量
Two vectors are equal if they have the same magnitude and the same direction. This means their components are identical. For example, (4, 1) = (4, 1).
若两个向量大小相同且方向相同，则它们相等，即对应分量完全相同。例如 (4, 1) = (4, 1)。
The negative of a vector has the same magnitude but the opposite direction. If a = (x, y), then:
负向量大小相同但方向相反。若 a = (x, y)，则：
−a = (−x, −y)
For instance, if a = (2, −5), then −a = (−2, 5). Geometrically, −a points in the exact opposite direction of a.
例如，若 a = (2, −5)，则 −a = (−2, 5)。在几何上，−a 与 a 的方向完全相反。
5. Addition and Subtraction of Vectors | 向量的加法与减法
To add vectors, add their corresponding components. If a = (3, 1) and b = (2, 5), then:
向量加法将对应分量相加。若 a = (3, 1)，b = (2, 5)，则：
a + b = (3 + 2, 1 + 5) = (5, 6)
Subtraction is done component-wise as well:
减法同样按分量进行：
a − b = (3 − 2, 1 − 5) = (1, −4)
Geometrically, when vectors are placed tip-to-tail, the vector from the tail of the first to the head of the second is the sum. This is called the triangle law of vector addition.
在几何上，当向量首尾相接时，从第一个向量的起点指向第二个向量终点的向量就是和向量，这称为向量加法的三角形法则。
6. Scalar Multiplication | 标量倍乘
Multiplying a vector by a scalar k multiplies each component by k. If a = (2, 3), then:
向量乘以标量 k，即每个分量都乘以 k。若 a = (2, 3)，则：
3a = (6, 9) and −2a = (−4, −6)
If k is positive, the direction of the vector is unchanged; if k is negative, the direction is reversed. Scalar multiplication also affects the magnitude: |ka| = k × |a|.
若 k 为正，向量方向不变；若 k 为负，方向反转。标量倍乘同样影响模长：|ka| = k × |a|。
This operation is the foundation for identifying parallel vectors, because parallel vectors are always scalar multiples of one another.
倍乘运算是判断平行向量的基础，因为平行向量总是互为标量倍。
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, (4, 8) and (2, 4) are parallel because (4, 8) = 2 × (2, 4).
若一个向量是另一个向量的标量倍，则二者平行。例如 (4, 8) 与 (2, 4) 平行，因为 (4, 8) = 2 × (2, 4)。
Points A, B, and C are collinear if the vectors AB and BC are parallel and they share the point B. In exam questions, you often must show that:
若向量 AB 与 BC 平行且它们共点 B，则 A、B、C 三点共线。考试中通常要求你证明：
AB = k × BC for some scalar k
This proves both parallelism and the common point, which together imply collinearity.
这同时证明了平行性与共点性，两者结合即可推出三点共线。
8. Position Vectors | 位置向量
A position vector describes the location of a point relative to the origin O. The point P(3, 4) has position vector p = (3, 4).
位置向量描述点相对于原点 O 的位置。点 P(3, 4) 的位置向量为 p = (3, 4)。
If A has position vector a and B has position vector b, then the vector from A to B is found by subtracting the initial position from the final position:
若 A 的位置向量为 a，B 的位置向量为 b，则从 A 指向 B 的向量等于终点位置向量减起点位置向量：
AB = b − a
For example, if A(1, 2) and B(4, 6), then AB = (4 − 1, 6 − 2) = (3, 4).
例如，A(1, 2) 与 B(4, 6)，则 AB = (4 − 1, 6 − 2) = (3, 4)。
9. Worked Example: Midpoint of a Triangle | 例题：三角形中点
Consider triangle OAB with O(0, 0), A(2, 3), and B(5, 1). Let M be the midpoint of AB. Find the position vector of M and the vector AB.
设三角形 OAB 中 O(0, 0)、A(2, 3)、B(5, 1)。M 为 AB 的中点。求 M 的位置向量以及向量 AB。
Solution: The midpoint M has coordinates ((2 + 5)/2, (3 + 1)/2) = (3.5, 2). Therefore the position vector is m = (3.5, 2).
解：中点 M 的坐标为 ((2 + 5)/2, (3 + 1)/2) = (3.5, 2)。因此位置向量 m = (3.5, 2)。
Now find AB = b − a = (5, 1) − (2, 3) = (3, −2). Notice that the midpoint formula can also be written in vector form:
再求 AB = b − a = (5, 1) − (2, 3) = (3, −2)。注意中点公式也可用向量表示：
m = (a + b) / 2 = ((2, 3) + (5, 1)) / 2 = (7, 4) / 2 = (3.5, 2)
This vector method is quick and avoids separate calculations for the x- and y-coordinates.
这种向量方法更快捷，无需分别计算横纵坐标。
10. Exam Tips and Common Errors | 考试技巧与常见错误
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Always draw a diagram for vector geometry questions. A clear sketch helps you see the direction of each vector.
向量几何题务必画图。清晰的草图有助于判断各向量的方向。
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When subtracting vectors, be careful with signs: a − b = a + (−b). A common error is reversing the order.
向量相减时注意符号：a − b = a + (−b)。常见错误是颠倒顺序。
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Check exactly what the question asks: a magnitude (a number) or a vector (with direction). Do not confuse |a| with a.
看清题目要求：求模长（数值）还是求向量（带方向）。不要把 |a| 与 a 混淆。
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When proving collinearity, always state the scalar multiple and mention the common point. Both parts are necessary.
证明共线时，必须说明标量倍数并指出共用的点。两者缺一不可。
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Use correct notation in your final answer. Write vectors in column form or bracket form, not as coordinates of a point unless asked.
最终答案使用正确记号。向量应写成列向量或括号形式，除非题目要求，否则不要写成点的坐标。
11. Practice Questions and Answers | 练习与答案
Test your understanding with these exam-style questions, then check the table below.
用以下真题风格题目检验你的理解，然后对照下表检查答案。
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Q1. If a = (3, −2) and b = (1, 4), find a + b, 2a − b, and |b|.
第1题：若 a = (3, −2)，b = (1, 4)，求 a + b、2a − b
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