📚 A-Level Maths: Vectors Revision Guide | A-Level 数学：向量 考点精讲
Vectors are fundamental in A-Level Mathematics, appearing in pure mathematics, mechanics, and further maths. They describe quantities with both magnitude and direction, such as displacement, velocity, and force. Understanding vector operations and their geometric interpretations is crucial for solving problems involving position, lines, and planes. This guide covers essential vector concepts, from basic notation to the scalar product and equations of lines, providing clear explanations and exam tips for students.
向量是 A-Level 数学的基础内容，广泛应用于纯数学、力学和进阶数学中。它们用于描述既有大小又有方向的量，例如位移、速度、力。掌握向量的运算及其几何意义，是解决位置、直线和平面相关问题的关键。本指南涵盖从基本符号到数量积、直线方程的核心向量概念，为学生提供清晰的解释和考试技巧。
1. What is a Vector? | 什么是向量？
A vector is a quantity that has both magnitude (size) and direction. In contrast, a scalar only has magnitude, such as mass or temperature. Vectors are used to represent displacement, velocity, acceleration, and forces. For instance, a velocity of 5 m/s east is a vector, while speed 5 m/s is a scalar. Vectors are geometrically represented as directed line segments, with an arrow indicating direction and length representing magnitude.
向量是既有大小又有方向的量。与之相对，标量只有大小，如质量或温度。向量用于表示位移、速度、加速度和力。例如，5 m/s 向东的速度是向量，而速率 5 m/s 是标量。向量在几何上可用有向线段表示，箭头指示方向，线段长度表示大小。
It is essential to distinguish vectors from scalars in exam questions. For example, distance is scalar, but displacement is a vector. Adding vectors follows geometric rules, not simple arithmetic.
在考试题目中区分向量和标量至关重要。例如，路程是标量，而位移是向量。向量的加法遵循几何法则，而非简单算术相加。
2. Vector Notation and Representation | 向量的符号与表示
Vectors can be written in several ways: as bold lowercase letters (a), with an arrow on top (→a), or as two points with an arrow indicating direction (AB). In component form, a vector in 2D is a = xi + yj, where i and j are unit vectors along the x- and y-axes. In 3D, we add a k component. Column vectors are also common, such as [x, y] as a vertical arrangement.
向量有多种书写方式：粗体小写字母 (a)、顶部带箭头 (→a)，或使用两点加箭头表示方向 (AB)。在分量形式中，二维向量 a = xi + yj，其中 i 和 j 分别是沿 x 轴和 y 轴的单位向量。三维中增加 k 分量。列向量也很常见，如竖排的 [x; y]。
For example, a vector from A(1,2) to B(4,6) can be written as AB = (4-1)i + (6-2)j = 3i + 4j, or as the column vector [3, 4]ᵀ. Its magnitude is √3²+4² = 5.
例如，从 A(1,2) 到 B(4,6) 的向量可写为 AB = (4-1)i + (6-2)j = 3i + 4j，或列向量 [3, 4]ᵀ。其模为 √3²+4² = 5。
3. Vector Addition and Subtraction | 向量的加法与减法
Vectors are added by placing them head-to-tail or by adding corresponding components. For vectors a = x₁i + y₁j and b = x₂i + y₂j, the sum a + b = (x₁+x₂)i + (y₁+y₂)j. Geometrically, this represents the resultant vector. Subtraction a - b is equivalent to a + (-b), and it gives the vector from the tip of b to the tip of a when placed tail-to-tail.
向量的加法可通过首尾相连的方法或分量相加进行。对于 a = x₁i + y₁j 和 b = x₂i + y₂j，和 a + b = (x₁+x₂)i + (y₁+y₂)j。几何上，这表示合向量。减法 a - b 等同于 a + (-b)，当起点对齐时，给出从 b 终点指向 a 终点的向量。
In the parallelogram law, if a and b are adjacent sides, a+b is the diagonal. Subtraction can also be understood via components: a - b = (x₁-x₂)i + (y₁-y₂)j.
在平行四边形法则中，若 a 和 b 为邻边，a+b 为对角线。减法也可通过分量理解：a - b = (x₁-x₂)i + (y₁-y₂)j。
4. Scalar Multiplication of Vectors | 向量的数乘
Multiplying a vector by a scalar λ scales its magnitude and can reverse direction if λ is negative. λa has magnitude |λ|·|a|; direction is the same as a if λ > 0, opposite if λ < 0. In components: λa = λ(xi + yj) = (λx)i + (λy)j.
向量乘以一个标量 λ 会缩放其大小，若 λ 为负还会反向。λa 的大小为 |λ|·|a|，方向与 a 相同当 λ > 0，相反当 λ < 0。分量为：λa = λ(xi + yj) = (λx)i + (λy)j。
Scalar multiplication helps check collinearity: two non-zero vectors a and b are parallel if one is a scalar multiple of the other, i.e., b = λa. If points A, B, C lie on a straight line, then AB and AC are parallel (AB = λ AC).
数乘可用于判断共线性：两个非零向量 a 和 b 平行，当且仅当存在标量 λ 使 b = λa。若点 A、B、C 在同一直线上，则向量 AB 和 AC 平行 (AB = λ AC)。
5. Magnitude of a Vector | 向量的模
The magnitude (or length) of a = xi + yj is |a| = √x² + y². In 3D, a = xi + yj + zk gives |a| = √x² + y² + z². This derives from Pythagoras' theorem. Magnitudes are always non-negative. A unit vector has magnitude 1.
向量 a = xi + yj 的模（长度）为 |a| = √x² + y²。三维中 a = xi + yj + zk 的模为 |a| = √x² + y² + z²。这源于勾股定理。模总是非负的。单位向量的模为 1。
To find the distance between points A(x₁,y₁) and B(x₂,y₂), use the magnitude of AB: |AB| = √((x₂-x₁)² + (y₂-y₁)²). Example: A(1,3) to B(4,7), distance = √3²+4² = 5.
求两点 A(x₁,y₁) 和 B(x₂,y₂) 的距离，使用 AB 的模：|AB| = √((x₂-x₁)² + (y₂-y₁)²)。例：A(1,3) 到 B(4,7)，距离 = √3²+4² = 5。
6. Unit Vectors | 单位向量
A unit vector has magnitude 1. The standard unit vectors in 2D are i = [1,0] and j = [0,1]. In 3D, we use i, j, k. Any non-zero vector a can be normalised to a unit vector in the same direction: â = a / |a|. This is widely applied in geometry and mechanics.
单位向量是模为 1 的向量。二维标准单位向量为 i = [1,0] 和 j = [0,1]。三维有 i、j、k。任何非零向量 a 可归一化为同方向的单位向量：â = a
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