📚 Vectors | 向量
Vectors are mathematical objects that describe quantities with both magnitude and direction, such as displacement, velocity, and force. In A-Level Edexcel Mathematics, vectors form a vital bridge between pure algebra and geometric reasoning, appearing in 2D and 3D coordinate systems, mechanics, and proofs. This revision guide breaks down every essential concept—from basic notation and component form to the scalar product and the vector equation of a line—so you can approach exam questions with confidence.
向量是描述既有大小又有方向的量的数学对象，例如位移、速度和力。在 A-Level Edexcel 数学中，向量是连接纯代数与几何推理的重要桥梁，出现在二维和三维坐标系、力学以及证明题中。本复习指南将拆解每一个核心概念——从基本记法和分量形式到标量积和直线的向量方程——帮助你有信心地应对考试题目。
1. Scalar vs Vector Quantities | 标量与向量
A scalar quantity has only magnitude (size), such as mass, temperature, or distance. A vector quantity has both magnitude and direction, such as displacement, velocity, or acceleration. The distinction is essential because vectors follow different rules of combination, including the need to consider direction when adding or subtracting them.
标量只有大小，例如质量、温度或距离。向量既有大小又有方向，例如位移、速度或加速度。这个区别很关键，因为向量在组合时遵循不同的规则，加减时必须考虑方向。
In diagrams, a vector is typically drawn as a directed line segment with an arrowhead showing direction. In written form, we represent vectors using bold typeface (e.g., a, AB) in print, or with an underlined letter in handwriting. For this article, bold symbols indicate vectors.
在示意图中，向量通常画成带箭头的有向线段，箭头表示方向。在书写和印刷中，我们用粗体字母表示向量（如 a、AB），手写时则在字母下加下划线。本文使用粗体符号表示向量。
2. Vector Notation and Component Form | 向量表示与分量形式
Vectors can be expressed in component form using the standard basis vectors i, j (in 2D) and i, j, k (in 3D). For example, a = 3i − 2j represents a vector with an x-component of 3 and a y-component of −2. In 3D, b = 4i + j + 5k has components along the x-, y-, and z-axes.
向量可以用标准基向量 i、j（二维）和 i、j、k（三维）的分量形式表示。例如 a = 3i − 2j 表示一个 x 分量为 3、y 分量为 −2 的向量。在三维中，b = 4i + j + 5k 拥有沿 x 轴、y 轴和 z 轴的分量。
Alternatively, we use column vector notation, which is especially common in mechanics and transformations:
a = [ 3, −2 ] (2D) or [ 4, 1, 5 ] (3D).
也可以用列向量表示法，这在力学和变换中尤其常见：
a = [ 3, −2 ]（二维）或 [ 4, 1, 5 ]（三维）。
The zero vector 0 has all components equal to zero and no specific direction. Equal vectors have identical magnitude and direction; their component forms must match exactly.
零向量 0 的所有分量均为零，没有特定的方向。相等的向量具有相同的大小和方向，它们的分量形式必须完全一致。
3. Magnitude and Direction | 大小与方向
The magnitude (or modulus) of a vector is its length. For a 2D vector a = xi + yj, the magnitude is given by Pythagoras' theorem:
|a| = √x² + y²
向量的大小（或模）是其长度。对于二维向量 a = xi + yj，大小由勾股定理给出：
|a| = √x² + y²
In 3D, for b = xi + yj + zk, we extend the formula:
|b| = √x² + y² + z²
在三维中，对于 b = xi + yj + zk，公式扩展为：
|b| = √x² + y² + z²
The direction of a 2D vector is usually described by the angle θ it makes with the positive x-axis, measured anticlockwise. From trigonometry:
tan θ = y / x (remember to check the quadrant)
二维向量的方向通常用它与 x 轴正方向所成的角 θ 来描述，按逆时针度量。根据三角学：
tan θ = y / x （注意判断象限）
In 3D, direction is defined by direction cosines or by stating the components, but A-Level Edexcel typically focuses on 2D direction angles for vectors and using the scalar product to find angles between vectors.
在三维中，方向由方向余弦或分量来确定，但 A-Level Edexcel 通常侧重于二维向量的方向角，以及利用标量积求向量之间的夹角。
4. Position Vectors and Displacement | 位置向量与位移向量
A position vector locates a point relative to the origin O. For a point A with coordinates (x, y), the position vector is OA = xi + yj. In 3D, if A = (x, y, z), then OA = xi + yj + zk.
位置向量确定一个点相对于原点 O 的位置。对于坐标为 (x, y) 的点 A，其位置向量为 OA = xi + yj。在三维中，若 A = (x, y, z)，则 OA = xi + yj + zk。
The vector that describes the movement from point A to point B is AB, and it is found by subtracting the position vectors:
AB = OB − OA = b − a
描述从点 A 到点 B 的运动的向量是 AB，可通过位置向量相减得到：
AB = OB − OA = b − a
This relation is fundamental for solving geometry problems involving midpoints, triangles, and parallelograms. The midpoint M of AB has position vector OM = ½ (a + b).
这个关系是解决涉及中点、三角形和平行四边形的几何问题的基础。线段 AB 的中点 M 的位置向量为 OM = ½ (a + b)。
5. Addition and Subtraction of Vectors | 向量的加法与减法
Vector addition follows the triangle law or the parallelogram law. Geometrically, to add a and b, place the tail of b at the tip of a; the resultant a + b goes from the tail of a to the tip of b. In component form, simply add corresponding components:
(x1i + y1j) + (x2i + y2j) = (x1 + x2)i + (y1 + y2)j
向量的加法遵循三角形法则或平行四边形法则。在几何上，要相加 a 和 b，把 b 的尾端放在 a 的尖端；合向量 a + b 从 a 的尾端指向 b 的尖端。在分量形式上，只需将对应分量相加：
(x1i + y1j) + (x2i + y2j) = (x1 + x2)i + (y1 + y2)j
Vector subtraction a − b is equivalent to adding the negative of b: a + (−b). The negative of a vector has the same magnitude but opposite direction. In components, subtract each coordinate: a − b = (x1 − x2)i + (y1 − y2)j.
向量减法 a − b 等价于加上 b 的负向量：a + (−b)。向量的负向量具有相同的大小但方向相反。在分量中，对各坐标做减法：a − b = (x1 − x2)i + (y1 − y2)j。
These operations are commutative for addition (a + b = b + a) and extend naturally to 3D by including the k-component.
加法满足交换律（a + b = b + a），这些运算可自然地推广到三维，只需包含 k 分量即可。
6. Scalar Multiplication and Parallel Vectors | 标量乘法与平行向量
When a vector is multiplied by a scalar (a real number), its magnitude is scaled by the absolute value of the scalar, and its direction either stays the same (if the scalar is positive) or reverses (if negative). If a = xi + yj, then ka = kxi + kyj.
当一个向量乘以一个标量（实数）时，其大小按该标量的绝对值缩放，方向保持不变（若标量为正）或反向（若标量为负）。若 a = xi + yj，则 ka = kxi + kyj。
Two vectors a and b are parallel if one is a scalar multiple of the other: b = λ a for some non-zero scalar λ. In component form, this means the ratios of corresponding components are equal (provided components are non-zero). Vectors a = 2i + 6j and b = i + 3j are parallel because a = 2b.
若一个向量是另一个向量的标量倍数，即 b = λ a（λ 为非零标量），则这两个向量平行。在分量形式下，这意味着对应分量的比值相等（当分量非零时）。向量 a = 2i + 6j 和 b = i + 3j 平行，因为 a = 2b。
Parallel vectors are crucial when dealing with vector equations of lines and when proving collinearity of points. Three points A, B, C are collinear if AB is parallel to BC (or AC).
在处理直线的向量方程以及证明点共线时，平行向量至关重要。若 AB 平行于 BC（或 AC），则 A、B、C 三点共线。
7. Unit Vectors | 单位向量
A unit vector has magnitude 1. The standard basis vectors i, j (and k in 3D) are unit vectors pointing along the coordinate axes. Any non-zero vector a can be converted into a unit vector in the same direction by dividing it by its magnitude:
â = a / |a|
单位向量的大小为 1。标准基向量 i、j（以及三维中的 k）是指向坐标轴方向的单位向量。任何非零向量 a 都可以通过除以其大小转化为同方向的单位向量：
â = a / |a|
For example, if a = 3i + 4j, then |a| = 5, so the unit vector is â = ⅗ 3i + ⅘ 4j = 0.6i + 0.8j. Unit vectors are often used to specify direction independently of magnitude.
例如，若 a = 3i + 4j，则 |a| = 5，故单位向量为 â = 0.6i + 0.8j。单位向量常用来在不考虑大小的情况下指定方向。
8. The Scalar Product (Dot Product) | 标量积（点积）
The scalar product of two vectors a and b, written a · b, is defined geometrically as:
a · b = |a||b| cos θ
where θ is the angle between the vectors when placed tail to tail. The result is a scalar, not a vector.
两个向量 a 和 b 的标量积，记作 a · b，其几何定义为：
a · b = |a||b| cos θ
其中 θ 是两向量尾尾相接时的夹角。结果是一个标量，而非向量。
In component form, for a = x1i + y1j + z1k and b = x2i + y2j + z2k, the scalar product is calculated as:
a · b = x1x2 + y1y2 + z1z2
在分量形式下，对于 a = x1i + y1j + z1k 和 b = x2i + y2j + z2k，标量积的计算公式为：
a · b = x1x2 + y1y2 + z1z2
Key properties: a · b = b · a (commutative); a · a = |a|2; and a · (b + c) = a · b + a · c (distributive law).
关键性质：a · b = b · a（交换律）；a · a = |a|2；以及 a · (b + c) = a · b + a · c（分配律）。
9. Angle Between Vectors and Perpendicularity | 向量间的夹角与垂直条件
Rearranging the scalar product formula gives the cosine of the angle θ between two non-zero vectors:
cos θ = (a · b) / (|a||b|)
重新整理标量积公式可得两非零向量夹角 θ 的余弦值：
cos θ = (a · b) / (|a||b|)
This allows us to find the angle whether the vectors are in 2D or 3D. For example, if a = 2i + 2j and b = 2i − j, then a · b = 4 − 2 = 2, |a| = √8, |b| = √5, so cos θ = 2 / (√8 √5) = 1/√10, giving θ ≈ 71.6°.
这样我们就可以求二维或三维向量的夹角。例如，若 a = 2i + 2j，b = 2i − j，则 a · b = 4 − 2 = 2，|a| = √8，|b| = √5，因此 cos θ = 2 / (√8 √5) = 1/√10，得 θ ≈ 71.6°。
A particularly important case is when a and b For non-zero vectors, perpendicularity implies θ = 90°, so cos θ = 0, giving:
a · b = 0
一个特别重要的情形是两向量互相垂直。对于非零向量，垂直意味着 θ = 90°，故 cos θ = 0，于是：
a · b = 0
This condition is widely used to prove that two lines are perpendicular, to find unknown components, or to construct coordinate geometry solutions.
这个条件广泛用于证明两条直线垂直、求解未知分量或构建坐标几何解法。
10. Vector Equation of a Straight Line | 直线的向量方程
A straight line in 2D or 3D can be expressed in vector form. Given a point A with position vector a on the line and a direction vector b parallel to the line, any point R on the line has position vector r given by:
r = a + t b
where t is a scalar parameter. This is the parametric vector equation of a line.
二维或三维中的直线可以用向量形式表示。给定直线上一点 A 的位置向量 a 以及平行于直线的一个方向向量 b，则直线上任意一点 R 的位置向量 r 为：
r = a + t b
其中 t 是标量参数。这就是直线的参数式向量方程。
For example, the line through (1, 3) parallel to 2i − j can be written as r = (i + 3j) + t(2i − j). By varying t, we generate all points on the line. In 3D, the idea is identical but with a k-component. The Cartesian equation can be obtained by equating components and eliminating t.
例如，过点 (1, 3) 且平行于向量 2i − j 的直线可写作 r = (i + 3j) + t(2i − j)。改变 t 的值，我们就能得到直线上所有点。三维中概念相同，只是增加了 k 分量。把各分量写成等式并消去 t，即可得到笛卡尔方程。
When solving intersection problems, you can set the position vectors of two lines equal (with different parameters, say t and s) and solve for the scalars. Check that the obtained values produce the same point.
求解直线交点问题时，可令两直线的位置向量相等（使用不同的参数，如 t 和 s），然后解出标量值，并检查所得的值是否确定同一点。
11. Parallel Vectors and Collinearity Proofs | 平行向量与共线性证明
To prove that three points A, B, C are collinear (lie on the same straight line), we show that the vectors AB and BC (or AC) are parallel. As seen earlier, this means one can be written as a scalar multiple of the other. For instance, if AB = 4i + 2j and BC = 2i + j, then AB = 2BC, so A, B, C are collinear.
要证明 A、B、C 三点共线（位于同一直线上），只需证明向量 AB
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