📚 GCSE Edexcel Maths: Vectors – Key Exam Points | GCSE Edexcel 数学：向量考点精讲
Vectors are a fundamental topic in GCSE Edexcel Maths, appearing on both Foundation and Higher tier papers. They describe quantities that have both magnitude and direction, unlike scalars which only have magnitude. Mastering vector notation, operations, and geometric applications is essential for success in the exam. This guide breaks down every key concept with clear examples, helping you tackle vector questions confidently.
向量是 GCSE Edexcel 数学中的核心内容，在基础级别和高级别试卷中都会出现。它们表示既有大小又有方向的量，这与只有大小的标量不同。掌握向量的表示法、运算和几何应用是考试成功的必要条件。本指南通过清晰示例逐一分解每个关键概念，帮助你自信应对向量题。
1. What Is a Vector? | 什么是向量？
A scalar quantity has only magnitude (size), for example speed 5 m/s, distance 10 km, mass 2 kg. A vector quantity has both magnitude and direction, such as velocity 5 m/s north, displacement 10 km east, or a force of 3 N acting to the right. In diagrams, vectors are drawn as arrows: the length represents the magnitude, and the arrowhead shows the direction. A vector from point A to point B is called displacement and written as AB.
标量仅有大小，例如速率 5 m/s、路程 10 km、质量 2 kg。向量既有大小又有方向，例如速度 5 m/s 向北、位移 10 km 向东，或向右的 3 N 力。在图中，向量画成箭头：长度代表大小，箭头表示方向。从点 A 到点 B 的向量称为位移，记作 AB。
2. Vector Notation | 向量的表示方法
Vectors can be written in several ways. In print, a single letter is often bold, e.g., a, b; in handwriting, it is common to underline the letter or place an arrow above it. In exam answers, you may use bold lowercase letters or two capital letters with an arrow, such as AB. The important thing is to be clear and consistent. Edexcel also accepts column vectors, which are very useful for calculations.
向量的书写方式有几种。印刷体常使用单个粗体字母，如 a、b；手写体通常给字母加下划线或加箭头。考试答案中，你可以使用粗体小写字母，或两个大写字母加箭头，如 AB。关键是要清晰且前后一致。Edexcel 也接受列向量的写法，在进行运算时尤为方便。
3. Column Vectors | 列向量
A column vector shows the horizontal (x) and vertical (y) components. For example, a vector that moves 3 units right and 1 unit up is written as ( 3
1 ). The top number is the change in the x-direction, the bottom number the change in the y-direction. The zero vector ( 0
0 ) has no length and no direction. The negative of a vector a = ( x
y ) is −a = ( −x
−y ); it has the same magnitude but opposite direction.
列向量展示水平（x）和竖直（y）分量。例如，向右移动 3 个单位、向上移动 1 个单位的向量写作 ( 3
1 )。上方数字是 x 方向的变化量，下方数字是 y 方向的变化量。零向量 ( 0
0 ) 没有长度也没有方向。向量 a = ( x
y ) 的负向量是 −a = ( −x
−y )；它的大小相同但方向相反。
4. Vector Addition | 向量加法
To add two vectors, simply add their corresponding components. If a = ( x₁
y₁ ) and b = ( x₂
y₂ ), then a + b = ( x₁ + x₂
y₁ + y₂ ). Geometrically, you can find the sum by placing the tail of the second vector at the head of the first vector; the resultant vector runs from the tail of the first to the head of the second. This is called the triangle law of vector addition.
两个向量相加，只需将对应分量相加。若 a = ( x₁
y₁ ) 且 b = ( x₂
y₂ )，则 a + b = ( x₁ + x₂
y₁ + y₂ )。几何上，你可以把第二个向量的起点放在第一个向量的终点处，和向量就是从第一个向量的起点指向第二个向量的终点的向量。这称为向量加法的三角形法则。
Addition is commutative: a + b = b + a. It is also associative: (a + b) + c = a + (b + c).
加法满足交换律：a + b = b + a。也满足结合律：(a + b) + c = a + (b + c)。
5. Vector Subtraction | 向量减法
Subtracting a vector is equivalent to adding its negative: a − b = a + (−b). In component form, a − b = ( x₁ − x₂
y₁ − y₂ ). Geometrically, a − b is the vector from the head of b to the head of a when both vectors start from the same point. For parallel vectors, subtraction can help find the vector between two points if you know their position vectors.
向量减法等价于加上其负向量：a − b = a + (−b)。用分量形式表示是 a − b = ( x₁ − x₂
y₁ − y₂ )。几何上，如果两个向量从同一点出发，a − b 就是从 b 的终点指向 a 的终点的向量。如果你知道点的位置向量，用减法可以方便地求出两点间的向量。
6. Scalar Multiplication | 标量乘法
When a vector is multiplied by a scalar k, each component is multiplied by k: ka = k( x
y ) = ( kx
ky ). The magnitude of the vector is scaled by |k|. If k > 0, the direction stays the same; if k < 0, the direction reverses. For example, −2a doubles the length and points opposite to a. Scalar multiplication is the foundation for parallel vectors and collinearity.
标量与向量相乘时，各分量都乘以该标量：ka = k( x
y ) = ( kx
ky )。向量的大小按 |k| 倍缩放。若 k > 0，方向不变；若 k < 0，方向相反。例如 −2a 长度变为 2 倍且方向与 a 相反。标量乘法是平行向量和共线性的基础。
7. Magnitude of a Vector | 向量的模长
The magnitude (or length) of a vector a = ( x
y ) is found using Pythagoras’ theorem: |a| = √(x² + y²). For example, the magnitude of ( 3
4 ) is √(3² + 4²) = √25 = 5. Magnitude is always a non-negative scalar. To find the distance between two points A and B, calculate |AB| = |b − a| where a and b are position vectors. A unit vector has magnitude 1; any non-zero vector can be converted to a unit vector by dividing by its magnitude.
向量 a = ( x
y ) 的模长（长度）可用勾股定理求得：|a| = √(x² + y²)。例如 ( 3
4 ) 的模长为 √(3² + 4²) = √25 = 5。模长始终是一个非负标量。为求两点 A 和 B 的距离，计算 |AB| = |b − a|，其中 a 和 b 是位置向量。单位向量的模长为 1；任意非零向量除以自身模长即可得到单位向量。
8. Parallel Vectors and Collinearity | 平行向量与共线
Two vectors are parallel if one is a scalar multiple of the other: a = kb for some non-zero scalar k. If k is positive, the vectors are in the same direction; if k is negative, they are opposite. For collinear points, you show that the vectors connecting them are parallel and share a common point. For example, points A, B, C are collinear if AB = kBC (or AC = kAB) and they meet at B (or A). Always check the common point — just showing parallel vectors is not enough.
若一个向量是另一个向量的标量倍数，则两者平行：存在非零标量 k 使得 a = kb。若 k 为正，方向相同；若 k 为负，方向相反。证明点共线时，需说明连接它们的向量互相平行且有一个公共点。例如，点 A、B、C 共线可通过 AB = kBC（或 AC = kAB）以及它们相交于 B（或 A）来证明。务必检查公共点——仅仅说明向量平行是不够的。
9. Solving Geometry Problems Using Vectors | 利用向量解决几何问题
Vectors provide a powerful tool for geometric proofs without relying on coordinates. Common tasks include: proving a quadrilateral is a parallelogram (show AB = DC or AB = b − a etc.), proving that a point is the midpoint of a line segment (show that the position vector of the midpoint is (a
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