📚 A-Level Mathematics: Basic Concepts of Vectors | A-Level 数学：向量的基本概念
This article introduces the fundamental concepts of vectors for Edexcel A-Level Mathematics. We will explore what vectors are, how to write them, and how to manipulate them, providing a solid foundation for future topics such as differentiation of vectors and three-dimensional geometry.
本文介绍 Edexcel A-Level 数学中向量的基本概念。我们将讨论向量的定义、表示方法及其基本运算，为后续学习向量微分与三维几何打下坚实基础。
1. What Is a Vector? | 什么是向量？
A vector is a quantity that has both magnitude (size) and direction. In contrast to a scalar, which only has a magnitude, a vector describes how to move from one point to another. For example, velocity is a vector because it includes speed and direction; temperature is a scalar because it only has a value.
向量是同时具有大小（模）和方向的量。与仅具有大小的标量不同，向量描述的是从一个点到另一个点的移动方式。例如，速度是向量，因为它包含速率和方向；温度是标量，因为它只有一个数值。
In two dimensions, a vector can be thought of as a directed straight line segment: the length of the segment represents the magnitude, and the arrow indicates the direction.
在二维平面中，向量可以看作一条有向线段：线段的长度代表模，箭头表示方向。
2. Notation | 向量表示法
There are several common ways to write a vector. In print, we often use bold lowercase letters such as v; when writing by hand, we put an arrow above the letter, e.g. v⃗ (but we will use the notation v here). Alternatively, we can write the displacement from point A to point B as overrightarrowAB.
向量的表示方法有多种。在印刷体中，我们常用粗体小写字母，如 v；手写时则在字母上方加箭头，例如 v⃗（本文仍用 v 表示）。另外，从点 A 到点 B 的位移可以写成 overrightarrowAB。
In component form, a two-dimensional vector in the coordinate plane can be written as a column vector:
在坐标平面中，二维向量可以用列向量表示：
v = (x, y)ᵀ
where x is the horizontal component and y is the vertical component. The same vector can be expressed in mathematical text as v = (x, y).
其中 x 是水平分量，y 是垂直分量。向量 v 也可写成 (x, y)。
3. Vector vs Scalar | 向量与标量
Scalars obey the ordinary rules of arithmetic, while vectors combine magnitude and direction. This distinction is crucial in physics and geometry: adding two scalars is simple, but adding two vectors depends on their directions.
标量遵循普通的算术运算规则，而向量则结合了大小和方向。这一区别在物理和几何中至关重要：两个标量相加很简单，但两个向量相加取决于它们的方向。
- Scalar examples: distance, speed, mass, time.
- 标量示例：距离、速率、质量、时间。
- Vector examples: displacement, velocity, force, acceleration.
- 向量示例：位移、速度、力、加速度。
For a displacement vector from A to B, the scalar distance is the magnitude of the vector, but the vector also carries the direction of motion.
对于从 A 到 B 的位移向量，标量距离是向量的模，但向量还包含运动方向。
4. Magnitude of a Vector | 向量的模
The magnitude (or modulus) of a vector v = (x, y) is written |v| and is given by the Pythagorean theorem:
向量 v = (x, y) 的模（或长度）记作 |v|，由勾股定理给出：
|v| = √x2 + y2
For a three-dimensional vector v = (x, y, z), the magnitude is |v| = √x² + y² + z².
对于三维向量 v = (x, y, z)，模为 |v| = √x² + y² + z²。
The magnitude is always non-negative and represents the length of the vector. For example, if v = (3, 4), then |v| = √3² + 4² = 5.
模永远非负，表示向量的长度。例如，若 v = (3, 4)，则 |v| = √3² + 4² = 5。
5. Unit Vectors and Basis Vectors | 单位向量与基向量
A unit vector is a vector of magnitude 1. For any non-zero vector v, the unit vector in the same direction is obtained by dividing the vector by its magnitude:
单位向量是模为 1 的向量。对于任何非零向量 v，同方向的单位向量可通过将向量除以它的模得到：
v̂ = v⁄|v|
In the Cartesian plane, the unit vectors along the positive x- and y-axes are usually denoted by i and j respectively. In three dimensions, we also use k for the z-axis direction. Any vector can be expressed in terms of these basis vectors:
在平面直角坐标系中，沿 x 轴和 y 轴正方向的单位向量通常记作 i 和 j。在三维空间中，z 轴方向用 k 表示。任何向量都可以用这些基向量表示：
v = xi + yj
where x and y are called the components of v.
其中 x 和 y 称为向量 v 的分量。
6. Position Vectors | 位置向量
A position vector defines the position of a point relative to the origin O. For a point P with coordinates (x, y), the position vector is overrightarrowOP = (x, y)ᵀ. It is often written as p.
位置向量定义了点相对于原点 O 的位置。对于坐标为 (x, y) 的点 P，其位置向量为 overrightarrowOP = (x, y)ᵀ，通常记作 p。
The displacement vector from point A to point B, where A has position vector a and B has position vector b, is:
从点 A 到点 B 的位移向量，其中 A 的位置向量为 a，B 的位置向量为 b，为：
overrightarrowAB = b - a
This is one of the most important vector concepts: it shows how to move from one point to another in terms of position vectors.
这是最重要的向量概念之一：它表明如何用位置向量来表示从一个点到另一个点的移动。
7. Vector Addition and Subtraction | 向量加减法
Vectors can be added component-wise. If u = (u₁, u₂) and v = (v₁, v₂), then:
向量可以按分量相加。若 u = (u₁, u₂) 且 v = (v₁, v₂)，则：
u + v = (u1 + v1, u2 + v2)
Subtraction is similar:
减法类似：
u - v = (u1 - v1, u2 - v2)
Geometrically, the sum of two vectors is found by placing the tail of the second vector at the head of the first; the resultant vector goes from the tail of the first to the head of the second. This is known as the triangle law.
在几何上，求两个向量的和时，将第二个向量的起点放在第一个向量的终点；和向量从第一个向量的起点指向第二个向量的终点。这就是三角形法则。
8. Scalar Multiplication | 数乘
When a vector is multiplied by a scalar k, its components are each multiplied by k:
当向量乘以标量 k 时，它的每个分量都乘以 k：
kv = (kx, ky)
If k > 0, the direction of the resulting vector is the same as the original; if k < 0, the direction is reversed. The magnitude becomes |k| × |v|.
若 k > 0，所得向量的方向与原向量相同；若 k < 0，方向相反。模变为 |k| × |v|。
For example, if v = (2, -1), then 3v = (6, -3) and -2v = (-4, 2). Notice that scalar multiplication can stretch, shrink, or reverse a vector.
例如，若 v = (2, -1)，则 3v = (6, -3)，-2v = (-4, 2)。注意数乘可以拉伸、缩短或反向向量。
9. Parallel Vectors | 平行向量
Two vectors a and b are parallel if one is a scalar multiple of the other. That is, a = kb for some real scalar k. If k > 0, they point in the same direction; if k < 0, they point in opposite directions.
两个向量 a 和 b 平行当且仅当其中一个向量是另一个的标量倍，即存在实数 k 使 a = kb。若 k > 0，它们同向；若 k < 0，它们反向。
₂y₁, i.e. their components are proportional. For three-dimensional vectors, the same rule applies: each component must be multiplied by the same factor.
在分量形式中，两个向量 (x₁, y₁) 和 (x₂, y₂) 平行如果 x₁y₂ = x₂y₁，即对应分量成比例。对于三维向量，同样适用：每一个分量都乘以同一因子。
This property is extremely useful in geometry for proving that points are collinear or that lines are parallel.
这一性质在几何中非常有用，可用于证明点共线或直线平行。
10. Simple Vector Geometry | 向量几何初步
Vectors provide an elegant tool for solving geometric problems. For example, the midpoint M of points A and B with position vectors a and b has position vector:
向量为解决几何问题提供了简洁的工具。例如，点 A 和 B 的位置向量分别为 a 和 b，其中点 M 的位置向量为：
m = a + b⁄2
Also, if a point P divides the line segment AB in the ratio m:n, then its position vector is given by:
同时，若点 P 按比例 m:n 分割线段 AB，则其位置向量为：
p = na + mb⁄m+n
These formulas arise directly from vector addition and scalar multiplication. They are frequently tested in A-Level exam questions involving diagrams with parallelograms, triangles, and collinear points.
这些公式直接由向量加法和数乘推导而来，是 A-Level 考试中经常考查的内容，常出现在涉及平行四边形、三角形和共线点的图形问题中。
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