📚 Vectors in Two and Three Dimensions | 二维与三维向量
Vectors are fundamental in mathematics because they capture both magnitude and direction, making them essential for modelling motion, forces, and geometric relationships. In two and three dimensions, vectors provide a powerful language to describe the position of points, the relative displacement between objects, and the way quantities combine. This article explores the core concepts of vector algebra, including components, magnitude, unit vectors, operations, the dot product, and how these ideas extend seamlessly from the plane into space. Whether you are working in the IB Mathematics: Analysis and Approaches or Applications and Interpretation courses, a solid understanding of vectors will strengthen your problem-solving toolkit across coordinate geometry, physics, and beyond.
向量在数学中至关重要，因为它们同时包含了大小和方向，这使其成为对运动、力和几何关系进行建模的基础。在二维和三维空间中，向量提供了一种强大的语言来描述点的位置、物体之间的相对位移以及物理量的合成方式。本文将探讨向量代数的核心概念，包括分量、模、单位向量、向量运算、点积，以及这些概念如何从平面无缝地推广到空间。无论你学习的是 IB 数学“分析与方法”还是“应用与解释”课程，扎实掌握向量都将增强你在坐标几何、物理等诸多领域中解决问题的能力。
1. Scalars and Vectors | 标量与向量
A scalar is a quantity that has only magnitude, such as length, mass, or temperature. A vector, by contrast, has both magnitude and direction. Typical examples in physics include velocity, acceleration, and force. In mathematics, vectors are often represented geometrically as directed line segments, where the length of the segment indicates the magnitude and the arrowhead indicates the direction. Notation-wise, a vector may be written as v, or between two points as AB for the vector from A to B.
标量是只有大小的量，例如长度、质量或温度。而向量则同时具有大小和方向。物理中常见的例子包括速度、加速度和力。在数学中，向量通常用有向线段几何地表示，线段的长度表示大小，箭头表示方向。在记法上，向量可以写成 v，也可以写成 AB 表示从点 A 到点 B 的向量。
2. Vector Notation and Representation | 向量的表示法
In component form, a vector in two dimensions is expressed as an ordered pair: v is the horizontal component and y is the vertical component. In three dimensions we write v ought of as the displacements along the coordinate axes. The zero vector is (0, 0) in 2D or (0, 0, 0) in 3D, and it has zero length with no specific direction. When writing vectors by hand, you may see an underline or an arrow above the letter, but in typeset text boldface is standard.
在分量形式中，二维向量可以表示为一个有序对：v = (x, y)。其中 x 是水平分量，y 是垂直分量。在三维空间中，我们写成 v = (x, y, z)。这些分量可以理解为沿坐标轴方向的位移。零向量在二维中是 (0, 0)，在三维中是 (0, 0, 0)，其长度为零，没有特定方向。手写向量时，可能会在字母上方加下划线或箭头，但在印刷体中通常使用粗体。
3. Magnitude of a Vector | 向量的模
The magnitude (or length) of a vector v = (x, y) is found using the distance formula derived from Pythagoras’ theorem:
|v| = √x² + y²
For a three-dimensional vector v = (x, y, z), the magnitude becomes:
|v| = √x² + y² + z²
The magnitude is always a non-negative scalar. A vector with magnitude 1 is called a unit vector.
向量 v = (x, y) 的模（或长度）利用由勾股定理导出的距离公式求得：|v| = √x² + y²。对于三维向量 v = (x, y, z)，其模为：|v| = √x² + y² + z²。模始终是一个非负的标量。模为 1 的向量称为单位向量。
4. Unit Vectors and Base Vectors | 单位向量与基向量
A unit vector in the direction of any non-zero vector v is obtained by dividing the vector by its magnitude: û = v / |v|. In the Cartesian plane, the standard base vectors are i = (1, 0) and j = (0, 1). Any 2D vector can be expressed as a linear combination: v = xi + yj. In three dimensions we add the base vector k = (0, 0, 1), so that v = xi + yj + zk. These base vectors are mutually perpendicular and have length one, forming an orthonormal basis.
任何非零向量 v 方向上的单位向量可以通过将向量除以其模得到：û = v / |v|。在笛卡儿平面中，标准基向量为 i = (1, 0) 和 j = (0, 1)。任何二维向量都可以表示为这两个向量的线性组合：v = xi + yj。在三维空间中，我们加入基向量 k = (0, 0, 1)，从而有 v = xi + yj + zk。这些基向量互相垂直且长度均为 1，构成一组标准正交基。
5. Position Vectors | 位置向量
A position vector is a vector that starts at the origin O and ends at a point P. If P has coordinates (x, y) in 2D, the position vector is OP = (x, y). For any two points A and B with position vectors a and b, the vector from A to B is given by AB = b − a. This is one of the most useful relationships in coordinate geometry, because it allows us to find displacements, midpoints, and distances purely algebraically.
位置向量是以原点 O 为起点、以点 P 为终点的向量。如果点 P 在二维空间的坐标为 (x, y)，则其位置向量为 OP = (x, y)。对于任意两点 A 和 B，若它们的位置向量分别为 a 和 b，则从 A 到 B 的向量为 AB = b − a。这是坐标几何中最有用的关系之一，因为它使我们能够纯粹通过代数运算求得位移、中点和距离。
6. Vector Addition and Subtraction | 向量的加法与减法
Vectors are added by placing them head-to-tail (triangle law) or by completing a parallelogram. In component form, addition is straightforward: (x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂). Subtraction is similarly performed component-wise: (x₁, y₁) − (x₂, y₂) = (x₁ − x₂, y₁ − y₂). These rules extend directly to three dimensions by including the z-components. Vector addition is commutative and associative, and the negative of a vector v is −v, which has the same magnitude but opposite direction.
向量的加法可以通过首尾相接（三角形法则）或作平行四边形来完成。在分量形式下，加法十分直接：(x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂)。减法同样逐分量进行：(x₁, y₁) − (x₂, y₂) = (x₁ − x₂, y₁ − y₂)。这些规则在加入 z 分量后可直接推广到三维。向量加法满足交换律和结合律，向量 v 的负向量是 −v，其大小相同但方向相反。
7. Scalar Multiplication | 标量乘法
When a vector is multiplied by a scalar k, each component is multiplied by k: k(x, y) = (kx, ky). Geometrically, this stretches or shrinks the vector by the factor |k|, and if k is negative the direction is reversed. Two non-zero vectors are parallel if one is a scalar multiple of the other. The concept of collinearity of points can be tested through vectors: points A, B and C are collinear if the vectors AB and BC (or AC) are parallel, i.e. if AB = λ BC for some scalar λ.
当向量与标量 k 相乘时，每个分量都乘以 k：k(x, y) = (kx, ky)。在几何上，这会将向量按因子 |k| 拉伸或缩短，如果 k 为负，则方向反向。如果两个非零向量中，一个是另一个的标量倍数，则它们平行。利用向量还可以检验点的共线性：如果向量 AB 与 BC（或 AC）平行，即存在某个标量 λ 使得 AB = λ BC，则点 A、B、C 三点共线。
8. The Dot Product (Scalar Product) | 点积（标量积）
The dot product of two vectors a and b is a scalar defined in two equivalent ways:
a · b = |a| |b| cos θ
where θ is the angle between them, and algebraically for 2D:
a · b = a₁b₁ + a₂b₂
In three dimensions, a · b for projecting one vector onto another and for measuring work done by a force in physics.
两个向量 a 和 b 的点积是一个标量，它有两种等价定义：a · b = |a| |b| cos θ，其中 θ 是两向量之间的夹角；在代数上，对于二维空间：a · b = a₁b₁ + a₂b₂。在三维空间中，a · b = a₁b₁ + a₂b₂ + a₃b₃。点积满足交换律且对加法可分配。当向量互相垂直（θ = 90°）时，点积为零，因为 cos 90° = 0。点积还用于将一个向量投影到另一个向量上，以及在物理中计算力所做的功。
9. Angle Between Two Vectors | 两向量的夹角
Rearranging the geometric definition gives a formula for the cosine of the angle:
cos θ = (a · b) / (|a| |b|)
proportion, or equivalently that the angle between them is 0° or 180°, making cos θ = ±1.
将几何定义重新排列，可以得到夹角的余弦公式：cos θ = (a · b) / (|a| |b|)。当向量已知时，这个公式在求三角形内角或直线间夹角时极为有用。检验向量是否垂直的一个快速方法就是看它们的点积是否为零。若要判断向量是否平行，我们通常检查分量是否成比例，也可以利用夹角为 0° 或 180° 时 cos θ = ±1 这一性质。
10. Vectors in Three Dimensions | 三维空间中的向量
Many concepts from two dimensions carry over naturally to three dimensions by adding a third component. The magnitude formula extends to |v| = √x² + y² + z². The base vectors i, j, k are used to write v = xi + yj + zk. The dot product remains a₁b₁ + a₂b₂ + a₃b₃, and the same cosine formula applies. In 3D, we also encounter the concept of a plane and the normal vector, but those are developed further in the IB curriculum under vector equations. For now, being confident with position vectors, distances, and the dot product in three dimensions lays a strong foundation.
在二维中引入的许多概念通过增加第三个分量自然地推广到三维。模的公式扩展为 |v| = √x² + y² + z²。基向量 i、j、k 用于将向量写作 v = xi + yj + zk。点积仍然为 a₁b₁ + a₂b₂ + a₃b₃，同样的夹角余弦公式也成立。在三维空间中我们还会接触平面和法向量的概念，但那些内容在 IB 课程中的向量方程部分将进一步展开。就目前而言，熟练掌握三维空间中的位置向量、距离以及点积将打下坚实的基础。
11. Applications in Geometry and Physics | 在几何与物理中的应用
Vectors are not abstract; they solve real problems. In geometry, the midpoint of AB has position vector (a + b n position. The dot product appears, for instance, in the formula for mechanical work: W = F · d, where F is a constant force vector and d is the displacement.
向量并非抽象的数学对象，它们能解决实际问题。在几何中，线段 AB 的中点的位置向量为 (a + b)/2。要证明一个四边形是平行四边形，只需证明其对边对应的向量相等。在物理中，作用在物体上的力可以作为向量相加来求合力；速度可以被分解为分量；位移向量则描述了位置的变化。点积出现在例如机械功的公式中：W = F · d，其中 F 是恒力向量，d 是位移向量。
12. Summary of Key Results | 重点公式总结
Below is a compact reference of the essential vector formulas for quick revision in both 2D and 3D contexts.
| Concept / 概念 | Formula (2D & 3D) / 公式 |
|---|---|
| Magnitude / 模 | |v| = √x² + y² ; √x² + y² + z² |
| Unit vector / 单位向量 | û = v / |v| |
| Displacement / 位移 | AB = b − a |
| Dot product / 点积 | a · b = a₁b₁ + a₂b₂ (+ a₃b₃) |
| Angle / 夹角 | cos θ = (a · b) / (|a| |b|) |
| Perpendicular / 垂直 | a · b = 0 |
| Parallel / 平行 | a = kb for some scalar k |
Remembering these relationships and practicing with past paper questions will build fluency in vector reasoning for IB examinations.
熟记这些关系并通过历年真题进行练习，将帮助你在 IB 考试中流利地运用向量进行推理。
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