📚 IB Math: Vectors – Key Concepts Summary | IB 数学：向量考点精讲
Vectors are a fundamental tool in IB Mathematics, appearing in both Analysis & Approaches (AA) and Applications & Interpretation (AI) courses, at Standard Level (SL) and Higher Level (HL). They link algebra with geometry, enabling us to describe positions, displacements, forces, and much more in 2D and 3D spaces. This article summarises the essential vector concepts, operations, and applications you need to master for the IB exams.
向量是 IB 数学中的基础工具，出现在分析与方法 (AA) 以及应用与解释 (AI) 课程中，涵盖标准级别 (SL) 和高级别 (HL)。向量将代数与几何联系起来，使我们能够描述二维和三维空间中的位置、位移、力等诸多物理量。本文总结了你需要掌握的 IB 考试核心向量概念、运算及应用。
1. Vector Basics: Magnitude, Direction, and Notation | 向量基础：模、方向与表示法
A vector is a quantity that has both magnitude and direction, such as displacement, velocity or force. Scalars have only magnitude, like mass or temperature.
向量是既有大小又有方向的量，例如位移、速度或力。标量只有大小，如质量或温度。
In IB, vectors are denoted by boldface letters (e.g., v) or with an arrow. In component form, a 2D vector is written as v = (x, y) or xi + yj, where i, j, (k in 3D) are unit vectors along the axes. The zero vector 0 has all components zero.
在 IB 中，向量用粗体字母（如 v）或带箭头表示。以分量形式表示时，二维向量写为 v = (x, y) 或 xi + yj，其中 i、j（三维中还有 k）是沿坐标轴方向的单位向量。零向量 0 的所有分量均为零。
Magnitude: |v| = √(x² + y²) in 2D, |v| = √(x² + y² + z²) in 3D
模（长度）：二维中 |v| = √(x² + y²)，三维中 |v| = √(x² + y² + z²)。
A unit vector has magnitude 1. Any non-zero vector can be converted to a unit vector in the same direction by dividing by its magnitude: û = v / |v|.
单位向量模长为 1。任何非零向量除以其模长即可得到同方向的单位向量：û = v / |v|。
2. Vector Operations: Addition, Subtraction, and Scalar Multiplication | 向量运算：加法、减法与数乘
Vectors are added geometrically by the triangle or parallelogram law. Algebraically, you simply add the corresponding components: if a = (a₁, a₂) and b = (b₁, b₂), then a + b = (a₁ + b₁, a₂ + b₂). The same applies in 3D.
向量相加在几何上遵循三角形或平行四边形法则。代数上只需将对应分量相加：若 a = (a₁, a₂)，b = (b₁, b₂)，则 a + b = (a₁ + b₁, a₂ + b₂)。三维同理。
Subtraction a – b is defined as a + (–b), meaning you reverse b and add. The components are subtracted: (a₁ – b₁, a₂ – b₂).
减法 a – b 定义为 a + (–b)，即将 b 反向后再相加。分量相减得 (a₁ – b₁, a₂ – b₂)。
Scalar multiplication λv changes the length by factor |λ|; if λ negative, the direction is reversed. Vectors a and b are parallel if b = λa for some scalar λ.
数乘 λv 将长度缩放 |λ| 倍；若 λ 为负，方向相反。若存在标量 λ 使得 b = λa，则两向量平行。
3. Scalar (Dot) Product: Definition and Properties | 标量积（点积）：定义与性质
The dot product combines two vectors to produce a scalar. It has two equivalent definitions: component form a · b = a₁b₁ + a₂b₂ + a₃b₃, and geometric form a · b = |a||b| cos θ, where θ is the angle between the vectors.
点积将两向量结合为一个标量。它有两种等价的定义：分量形式 a · b = a₁b₁ + a₂b₂ + a₃b₃，以及几何形式 a · b = |a||b| cos θ，其中 θ 为两向量的夹角。
Key properties are summarised below (encompassing SL and HL):
主要性质总结如下（涵盖 SL 与 HL）：
| Property | 中文 |
|---|---|
| a · b = b · a (commutative) | 交换律 |
| a · (b + c) = a · b + a · c (distributive) | 分配律 |
| a · a = |a|² | 自身点积等于模长的平方 |
| a · b = 0 ⇔ a ⟂ b (nonzero vectors) | 点积为零等价于两非零向量垂直 |
4. Applications of Dot Product: Angle and Projection | 点积的应用：夹角与投影
The angle between vectors a and b is found by rearranging the dot product formula: cos θ = (a · b) / (|a||b|). This works in both 2D and 3D and is fundamental for solving geometric problems.
通过重组点积公式可求得向量 a 与 b 的夹角：cos θ = (a · b) / (|a||b|)。该公式在二维与三维中均适用，是几何问题的基础。
The scalar projection of a onto b is compb a = (a · b) / |b|, representing the length of the ‘shadow’ of a along the direction of b. The vector projection is projb a = ((a · b) / |b|²) b.
a 在 b 上的标量投影为 compb a = (a · b) / |b|，表示 a 沿 b 方向的“影子”长度。向量投影为 projb a = ((a · b) / |b|²) b。
These concepts are used to resolve forces, find the foot of a perpendicular, and compute shortest distances. Perpendicularity checks with a · b = 0 are particularly important in vector geometry.
这些概念用于力的分解、求垂足以及计算最短距离。利用 a · b = 0 检验垂直性在向量几何中尤为重要。
5. Vector (Cross) Product – HL Only | 向量积（叉积）– 仅HL
In HL, the cross product a × b yields a vector perpendicular to both a and b, with direction determined by the right-hand rule. Its magnitude |a × b| = |a||b| sin θ gives the area of the parallelogram spanned by the vectors.
在 HL 中，叉积 a × b 得到一个垂直于 a 与 b 的向量，方向由右手定则确定。其大小 |a × b| = |a||b| sin θ，等于以两向量为边的平行四边形面积。
Component formula: a × b = (a₂b₃ – a₃b₂)i – (a₁b₃ – a₃b₁)j + (a₁b₂ – a₂b₁)k
分量计算公式如上行所示，也可用行列式记忆。注意叉积是反交换的：a × b = –(b × a)。叉积在求平面法向量、三角形面积以及四面体体积中不可或缺。
6. Vector Equation of a Line in 2D and 3D | 二维与三维中直线的向量方程
A straight line can be expressed as r = a + td, where a is the position vector of a fixed point on the line, d is a direction vector, and t is a real parameter. This form works identically in 2D and 3D.
一条直线可表示为 r = a + td，其中 a 是直线上一定点的位置向量，d 是方向向量，t 为实参数。该形式在二维与
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