📚 Working with Vectors | 向量运算
Vectors lie at the heart of A-Level Mathematics, underpinning topics in pure maths, mechanics and beyond. This article walks you through the fundamental operations you will meet when working with vectors, from basic notation to solving geometric problems.
向量是 A-Level 数学的核心，支撑着纯数学、力学等多个领域。本文将带你走过向量运算中会遇到的各项基本操作，从基础表示法到解决几何问题。
1. What is a Vector? | 什么是向量？
A vector is a quantity that possesses both magnitude (size) and direction. Common examples include displacement, velocity and force. In contrast, a scalar has magnitude only – think of speed, distance or mass.
向量是既有大小又有方向的量，常见的例子有位移、速度和力。相反，标量只有大小，比如速率、路程和质量。
Vectors can be represented geometrically by arrows, where the length indicates magnitude and the arrowhead shows direction. Two vectors are equal if they have the same magnitude and direction, regardless of their starting points.
向量可以用箭头几何表示，箭头的长度表示大小，箭头指向表示方向。两个向量只要大小和方向相同就相等，与起点位置无关。
You will see vectors written in bold (e.g. a) or with an arrow above (→a). In handwriting, underlining or an arrow is used.
你常见向量以粗体书写（如 a）或上方加箭头（→a）。手写时通常用下划线或箭头表示。
2. Notation: Column Vectors and i, j Form | 表示法：列向量和 i, j 形式
In two dimensions, a vector can be expressed as a column vector. For instance, a vector moving 3 units right and 4 units up is written as (3, 4) in column form or, more commonly, with a vertical arrangement: a = (34).
在二维中，向量可以写成列向量的形式。比如，向右 3 个单位、向上 4 个单位的向量，可表示成列向量：a = (34)。
The same vector can be written using the standard basis vectors i and j: a = 3i + 4j. Here i represents one unit in the positive x-direction, and j one unit in the positive y-direction.
同样的向量可以用标准基向量 i 和 j 表示：a = 3i + 4j。这里 i 表示 x 轴正方向的一个单位，j 表示 y 轴正方向的一个单位。
In three dimensions we add k for the z-direction, so v = 2i - 5j + k would have components (2, -5, 1).
在三维空间中，我们增加 k 表示 z 轴方向，因此 v = 2i - 5j + k 的分量为 (2, -5, 1)。
3. Adding Vectors | 向量加法
To add two vectors, simply add their corresponding components. If a = a1i + a2j and b = b1i + b2j, then a + b = (a1 + b1)i + (a2 + b2)j. The same rule applies to column vectors by adding row-wise.
向量相加，只需将对应的分量相加。若 a = a1i + a2j，b = b1i + b2j，则 a + b = (a1 + b1)i + (a2 + b2)j。列向量相加同理，逐行相加。
Geometrically, vector addition follows the triangle law: place the tail of the second vector at the head of the first. The resultant goes from the tail of the first to the head of the second. Equivalently, the parallelogram law can be used.
几何上，向量加法遵循三角形法则：将第二个向量的起点放在第一个向量的终点，合向量从第一个起点指向第二个终点。等价地，也可以用平行四边形法则。
Addition is commutative, so a + b = b + a. This is useful when rearranging vector equations.
加法满足交换律，即 a + b = b + a，这在整理向量方程时很有用。
4. Subtracting Vectors | 向量减法
Subtracting a vector is equivalent to adding its negative. The negative of a vector keeps the same magnitude but points in the opposite direction. If b = 3i + 2j, then -b = -3i - 2j.
向量减法等同于加上它的负向量。负向量大小不变但方向相反。若 b = 3i + 2j，则 -b = -3i - 2j。
To compute a - b, subtract the components: (a1 - b1)i + (a2 - b2)j. This is particularly handy when determining a vector joining two points; for example, the vector from A to B is AB = OB - OA.
计算 a - b 时，逐分量相减：(a1 - b1)i + (a2 - b2)j。这在确定两点之间的向量时特别有用；例如从 A 到 B 的向量为 AB = OB - OA。
Geometrically, a - b can be drawn by placing the tails of a and b together, and drawing the vector from the head of b to the head of a.
几何上，a - b 可以把 a 和 b 的起点重叠，然后从 b 的终点指向 a 的终点。
5. Scalar Multiplication | 标量乘法
Multiplying a vector by a scalar (a real number) changes its magnitude but not its direction, unless the scalar is negative, which also reverses direction. If v = 2i - 3j and λ = 4, then λv = 8i - 12j.
向量乘以标量（一个实数）会改变其大小但不改变方向，除非标量为负，此时方向反转。若 v = 2i - 3j 且 λ = 4，则 λv = 8i - 12j。
Scalar multiplication is distributive: λ(a + b) = λa + λb. This property is essential when simplifying vector expressions or proving collinearity.
标量乘法满足分配律：λ(a + b) = λa + λb。这一性质在化简向量表达式或证明共线时至关重要。
If λ > 1, the vector gets longer; if 0 < λ < 1, it gets shorter. A zero scalar gives the zero vector, 0.
当 λ > 1 时，向量变长；0 < λ < 1 时，向量缩短。标量为零得到零向量 0。
6. Magnitude of a Vector | 向量的大小（模）
The magnitude (or modulus) of a vector is its length, found using Pythagoras' theorem. For a = xi + yj, the magnitude is |a| = √x² + y². For example, if a = 3i + 4j, then |a| = √9 + 16 = 5.
向量的大小（或模）是其长度，可以通过勾股定理求得。对于 a = xi + yj，模为 |a| = √x² + y²。例如 a = 3i + 4j，则 |a| = √9 + 16 = 5。
In three dimensions, |v| = √x² + y² + z² where v = xi + yj + zk. Magnitude is always non-negative.
在三维中，|v| = √x² + y² + z²，其中 v = xi + yj + zk。模总是非负值。
The magnitude of a velocity vector gives speed; the magnitude of a displacement vector gives distance.
速度向量的模给出速率；位移向量的模给出距离。
7. Unit Vectors | 单位向量
A unit vector has a magnitude of 1 and simply indicates direction. Any non‑zero vector can be converted into a unit vector by dividing by its magnitude: â = a / |a|. The hat symbol ( ̂ ) is often used.
单位向量的大小为 1，仅用来表示方向。任何非零向量除以其模，即可转化为单位向量：â = a / |a|。常用“帽子”符号（ ̂ ）表示。
For example, if a = 3i + 4j, then |a| = 5 and â = (3/5)i + (4/5)j. In mechanics, unit vectors are useful for describing forces in specified directions.
例如，若 a = 3i + 4j，则 |a| = 5，â = (3/5)i + (4/5)j。力学中，单位向量常用于描述特定方向上的力。
The base vectors i, j and k are themselves unit vectors pointing along the Cartesian axes.
基向量 i、j、k 本身就是沿直角坐标轴方向的单位向量。
8. Position Vectors | 位置向量
A position vector gives the location of a point relative to a fixed origin O. For point A with coordinates (x, y), its position vector is OA = xi + yj. This ties together coordinate geometry and vectors.
位置向量是描述点相对于固定原点 O 的位置的向量。对于坐标为 (x, y) 的点 A，其位置向量为 OA = xi + yj。这便将坐标几何和向量联系在了一起。
The vector from point A to point B is always AB = OB - OA, regardless of where the origin is. This is one of the most powerful tools for solving geometry problems with vectors.
无论原点在哪里，从点 A 到点 B 的向量始终为 AB = OB - OA。这是用向量解决几何问题最强有力的工具之一。
If three points A, B and C are collinear, the vectors AB and AC are scalar multiples of each other. Similarly, the midpoint M of AB has position vector OM = (OA + OB)/2.
若 A、B、C 三点共线，则向量 AB 与 AC 互为标量倍数。同理，AB 的中点 M 的位置向量为 OM = (OA + OB)/2。
9. Solving Geometric Problems | 解决几何问题
Many A-Level questions ask you to use vectors to prove relationships such as midpoints, collinearity and section ratios. The key is to express everything in terms of known position vectors and use scalar multiples.
许多 A-Level 题目要求用向量证明中点、共线、分割比等关系。关键在于将所有东西用已知的位置向量表示出来，并利用标量倍数的性质。
For instance, to prove that a point P divides AB in the ratio λ:μ, you can show that OP = (μOA + λOB)/(λ+μ). This formula is a direct consequence of the section formula.
例如，要证明点 P 以 λ:μ 的比例分割 AB，可以证明 OP = (μOA + λOB)/(λ+μ)。该公式是定比分点公式的直接结果。
Parallel vectors are scalar multiples of each other, and a zero vector results when two equal but opposite vectors are added. These simple ideas form the backbone of vector proofs.
平行向量互为标量倍数，两个大小相等方向相反的向量相加得零向量。这些简单的概念构成了向量证明的骨架。
Always draw a clear diagram and label the known vectors. Working methodically from the given information will usually break the problem down into simple component additions and subtractions.
务必画出清晰的示意图并标出已知向量。从已知信息有条理地推导，通常能将问题分解为简单的分量加减运算。
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