📚 Working with Vectors | 向量运算与几何应用
Vectors are one of the highest-value topics in Edexcel A-Level Mathematics because they connect algebra, trigonometry and geometry in a single toolkit. A vector describes a quantity that has both magnitude and direction, such as displacement, velocity, acceleration and force, whereas a scalar has magnitude only. This revision guide works through the full skill chain: notation and representation, magnitude and direction, unit vectors, addition and subtraction, scalar multiplication and parallel vectors, position vectors, ratio division and collinearity, vector equations of lines, intersections and skew lines, the scalar product, and finally geometric applications with exam technique.
向量是 Edexcel A-Level 数学中分值最高的专题之一，因为它把代数、三角和几何统一到同一套工具中。向量描述既有大小又有方向的量，例如位移、速度、加速度和力，而标量只有大小。本复习指南按完整技能链展开：符号与表示、大小与方向、单位向量、加法与减法、数乘与平行向量、位置向量、定比分点与共线、直线的向量方程、交点与异面直线、数量积，最后是几何应用与考试技巧。
1. Vector Notation and Representation | 向量符号与表示
A vector is normally printed in bold type, for example a, or with an arrow above the letter, for example the vector from A to B is written with an arrow over AB. In handwriting, an underline under the letter is the accepted convention. The magnitude, or modulus, of a vector is written |a| and is always a scalar, never negative.
向量通常印刷为粗体，例如 a，或在字母上方加箭头，例如从 A 到 B 的向量写作 AB 上方带箭头。手写时公认的规范是在字母下方加下划线。向量的大小（模）写作 |a|，它始终是标量，且不为负。
In two dimensions a vector can be written as a column vector with an x-component on top and a y-component below, or in component form using the standard basis vectors i and j. The vector (3, 4) as a column is exactly the same vector as 3i + 4j. Both forms are accepted in exams, but you must be able to convert quickly and accurately between them.
在二维中，向量可以写成列向量，上方是 x 分量，下方是 y 分量；也可以用标准基向量 i 和 j 写成分量形式。列向量 (3, 4) 与 3i + 4j 是完全相同的向量。两种形式在考试中都被接受，但你必须能够快速准确地相互转换。
In three dimensions we add a third basis vector k pointing along the positive z-axis, so a general vector is a = xi + yj + zk. Negative components simply mean the vector points in the negative coordinate direction. Two vectors are equal if and only if every corresponding component is equal, which is why component comparison is the standard method for solving vector equations.
在三维中我们增加第三个基向量 k，沿 z 轴正方向，因此一般向量为 a = xi + yj + zk。负分量只表示向量指向坐标轴的负方向。两个向量相等，当且仅当每一个对应分量都相等，这正是"比较分量"成为解向量方程标准方法的原因。
a = xi + yj = (x, y)
The zero vector 0 has magnitude zero and no defined direction. It appears naturally in equilibrium problems: if three forces act on a particle in equilibrium, their vector sum equals 0.
零向量 0 的大小为零，方向未定义。它自然出现在平衡问题中：若三个力作用于处于平衡的质点，它们的向量和等于 0。
2. Magnitude and Direction | 向量的大小与方向
The magnitude of a two-dimensional vector comes directly from Pythagoras' theorem. If a = xi + yj, then |a| = √x² + y². For a three-dimensional vector a = xi + yj + zk, the same idea extends to |a| = √x² + y² + z².
二维向量的大小直接来自勾股定理。若 a = xi + yj，则 |a| = √x² + y²。对三维向量 a = xi + yj + zk，同样的思路推广为 |a| = √x² + y² + z²。
|a| = √x² + y² + z²
0°. The starting point is tan θ = y ÷ x, but this alone is not enough, because tangent repeats every 180°. You must always check which quadrant the vector lies in and adjust the angle, otherwise you will lose the accuracy mark.
二维向量的方向通常表示为从正 x 轴逆时针量度的角 θ，范围为 0° ≤ θ < 360°。出发点是 tan θ = y ÷ x，但仅有这一步并不够，因为正切每 180° 重复一次。你必须判断向量所在象限并调整角度，否则会丢掉准确性分。
For example, if a = 3i + 4j, then |a| = √9 + 16 = √25 = 5 and θ = arctan(4 ÷ 3) = 53.1° to one decimal place. If b = −2i + 3j, then |b| = √13 se the vector is in the second quadrant, θ = 180° − arctan(3 ÷ 2) = 180° − 56.3° = 123.7°.
例如，若 a = 3i + 4j，则 |a| = √9 + 16 = √25 = 5，θ = arctan(4 ÷ 3) = 53.1°（保留一位小数）。若 b = −2i + 3j，则 |b| = √13 ≈ 3.61，由于该向量位于第二象限，θ = 180° − arctan(3 ÷ 2) = 180° − 56.3° = 123.7°。
Some questions ask for the direction as a bearing rather than as an angle from the x-axis. In that case measure clockwise from due north, and remember that a vector pointing north-east with equal components has a bearing of 045°. Always reread the question to see which convention is required.
有些题目要求用方位角而不是与 x 轴的夹角来表示方向。此时要从正北方向顺时针量度，并记住分量的北向与东向相等时方位角为 045°。务必重读题目，确认要求使用哪种约定。
3. Unit Vectors | 单位向量
A unit vector is a vector of magnitude 1. Every non-zero vector a has a unit vector in the same direction, found by dividing the vector by its own magnitude. This process is called normalising the vector.
单位向量是大小为 1 的向量。任何非零向量 a 都有一个与之同方向的单位向量，求法是把该向量除以其自身的大小。这个过程称为向量的单位化。
â = a ÷ |a|
For a = 3i + 4j with |a| = 5, the unit vector is â = (3 ÷ 5)i + (4 ÷ 5)j = 0.6i + 0.8j. Checking: √0.6² + 0.8² = √0.36 + 0.64 heck like this is worth doing in the exam because it catches arithmetic slips immediately.
对 a = 3i + 4j，|a| = 5，单位向量为 â = (3 ÷ 5)i + (4 ÷ 5)j = 0.6i + 0.8j。检验：√0.6² + 0.8² = √0.36 + 0.64 = 1，符合要求。这种快速检验在考试中很值得做，因为它能立即发现算术失误。
Unit vectors are useful when you need a vector of a given length in a given direction. To build a vector of magnitude k in the direction of a, simply multiply the unit vector by k: the answer is k â. For example, a vector of length 10 in the direction of 3i + 4j is 10(0.6i + 0.8j) = 6i + 8j.
当你需要在给定方向上得到给定长度的向量时，单位向量很有用。要构造方向与 a 相同、大小为 k 的向量，只需把单位向量乘以 k：答案为 k â。例如，方向与 3i + 4j 相同、长度为 10 的向量是 10(0.6i + 0.8j) = 6i + 8j。
Note that i, j and k are themselves unit vectors, and they are mutually perpendicular. This is the reason the component form is so convenient for both magnitude and the scalar product.
注意 i、j、k 本身就是单位向量，而且彼此垂直。这正是分量形式在求大小和数量积时都非常方便的原因。
4. Vector Addition and Subtraction | 向量加法与减法
Geometrically, vector addition follows the triangle law: to find a + b, draw a, then draw b starting from the end of a; the resultant runs from the start of a to the end of b. The parallelogram law gives the same answer and is often clearer in diagrams involving two forces from the same point.
在几何上，向量加法遵循三角形法则：求 a + b 时，先画 a，再从 a 的终点开始画 b；合向量从 a 的起点指向 b 的终点。平行四边形法则给出相同结果，在涉及同一点出发的两个力的图中往往更清晰。
Algebraically, addition is done component by component. If a = 3i + 4j and b = i − 2j, then a + b = 4i + 2j. Subtraction is addition of the negative vector, so a − b = 3i + 4j − (i − 2j) = 2i + 6j. Combined operations work in the usual algebraic way: 2a − 3b = (6i + 8j) − (3i − 6j) = 3i + 14j.
在代数上，加法按分量逐个进行。若 a = 3i + 4j，b = i − 2j，则 a + b = 4i + 2j。减法是加上相反向量，因此 a − b = 3i + 4j − (i − 2j) = 2i + 6j。组合运算按通常代数方式进行：2a − 3b = (6i + 8j) − (3i − 6j) = 3i + 14j。
the sum of the two legs, so AB + BC = AC. Rearranged, this gives the vector from A to B as b − a when the position vectors of A and B are a and b respectively. Reversing a vector changes its sign: BA = −AB.
最有用的结论是位移法则：若能先从 A 到 B，再从 B 到 C，则从 A 到 C 的向量等于两段之和，即 AB + BC = AC。整理后可得：当 A、B 的位置向量分别为 a、b 时，从 A 到 B 的向量为 b − a。反向的向量改变符号：BA = −AB。
Addition is commutative (a + b = b + a) and associative ((a + b) + c = a + (b + c)), which means you can combine several vectors in any convenient order.
加法满足交换律（a + b = b + a）和结合律（(a + b) + c = a + (b + c)），这意味着你可以按任何方便的次序合并多个向量。
5. Scalar Multiplication and Parallel Vectors | 数乘与平行向量
Multiplying a vector by a scalar λ produces a vector λa whose length is |λ| |a| sult is the zero vector. Multiplication by a scalar is done component by component, so λ(xi + yj) = λxi + λyj.
把向量乘以标量 λ 得到向量 λa，其长度为 |λ| |a|。若 λ > 0，方向不变；若 λ < 0，方向相反；若 λ = 0，结果为零向量。数乘按分量逐个进行，所以 λ(xi + yj) = λxi + λyj。
|λa| = |λ| × |a|
Two non-zero vectors are parallel exactly when one is a scalar multiple of the other: a = λb for some non-zero λ. In column form this means the components are in the same ratio, so the determinant test x₁y₂ − x₂
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