📚 Vectors in the Plane | 平面向量
Vectors are one of the most versatile tools in IB Mathematics, bridging geometry, physics, and algebra. A vector in the plane captures both magnitude and direction, allowing us to model displacement, velocity, and forces with precision. This article unpacks the core concepts of two‑dimensional vectors — from notation and basic operations to the dot product and the vector equation of a line — equipping you for success in Analysis & Approaches as well as Applications & Interpretation.
向量是 IB 数学中最通用的工具之一，搭起了几何、物理与代数之间的桥梁。平面中的向量既包含大小也包含方向，使我们能够精确地描述位移、速度和力。本文全面梳理二维向量的核心概念——从表示法和基本运算到点积与直线的向量方程——助力你在分析与方法、应用与解释两门课程中取得优异成绩。
1. What is a Vector? | 什么是向量？
A scalar is a quantity that has only magnitude, such as mass (5 kg) or temperature (30 °C). A vector, by contrast, possesses both magnitude and direction — think of a wind blowing at 15 km/h towards the north‑east. In the plane, a vector can be visualised as a directed line segment, where the length corresponds to the magnitude and the arrowhead indicates the direction.
标量是只有大小的量，如质量（5 kg）或温度（30 °C）。向量则同时具有大小和方向——想象一下以 15 km/h 向东北吹的风。在平面中，向量可以看作有向线段，线段的长度表示大小，箭头指向表示方向。
2. Vector Notation and Representation | 向量的表示法与表示形式
We usually denote a vector by a bold lowercase letter, such as v, or by an arrow above the letter, v⃗. In component form, a vector in the plane is written as v = x i + y j or as a column matrix (x, y)ᵀ, where x and y are the horizontal and vertical displacements. Geometrically, a vector can be placed anywhere; if it starts at the origin, its terminal point gives the components directly.
我们通常用一个粗体小写字母表示向量，如 v，或在字母上方加箭头——v⃗。在分量形式中，平面内的向量记为 v = x i + y j，或写成列矩阵 (x, y)ᵀ，其中 x 和 y 分别为水平位移与竖直位移。几何上，向量可以画在任意位置；若其起点为原点，则终点坐标直接给出分量。
3. Magnitude and Direction | 模与方向
The magnitude (or length) of a vector v = x i + y j is found using Pythagoras’ theorem:
|v| = √x² + y²
The direction is often described by the angle θ measured anticlockwise from the positive x‑ awareness). Mastering magnitude and direction is essential for resolving forces and calculating relative velocities.
向量 v = x i + y j 的模（长度）由勾股定理给出：
|v| = √x² + y²
方向通常用从正 x 轴逆时针测量的角度 θ 描述，满足 tan θ = y/x（需注意所在象限）。熟练掌握模与方向对于力的分解和相对速度的计算至关重要。
4. Vector Addition and Subtraction | 向量加法与减法
To add two vectors geometrically, place the tail of the second at the tip of the first; the resultant vector runs from the first tail to the second tip (triangle law). The parallelogram law places both tails together and takes the diagonal. In component terms, (x₁ i + y₁ j) + (x₂ i + y₂ j) = (x₁ + x₂) i + (y₁ + y₂) j. Subtraction a – b is equivalent to a + (–b), simply reversing the direction of b.
几何上，向量加法将第二个向量的尾端置于第一个向量的尖端，合向量从第一个尾端指向第二个尖端（三角形法则）。平行四边形法则将两者尾端重合，取对角线。在分量运算中，(x₁ i + y₁ j) + (x₂ i + y₂ j) = (x₁ + x₂) i + (y₁ + y₂) j。减法 a – b 等同于 a + (–b)，即将 b 的方向反转。
5. Scalar Multiplication | 数乘向量
Multiplying a vector by a scalar k stretches or shrinks it: k (x i + y j) = (kx) i + (ky) j. If k > 0, the direction stays the same; if k < 0, the direction reverses. Two non‑zero vectors are parallel (collinear) if one is a scalar multiple of the other, a property widely used to prove concurrency and collinearity in geometry problems.
向量乘以标量 k 会伸长或缩短：k (x i + y j) = (kx) i + (ky) j。若 k > 0，方向不变；若 k < 0，方向相反。两个非零向量平行（共线）当且仅当一个是另一个的标量倍数，这一性质广泛用于几何中证明共线或共点问题。
6. Position Vectors | 位置向量
Fix an origin O. The position vector of a point A is the vector OA that starts at O and ends at A. If A has coordinates (x, y), its position vector is a = x i + y j. The vector joining two points A and B can be expressed as AB = OB – OA. Position vectors bring a powerful algebraic structure to coordinate geometry.
选定原点 O。点 A 的位置向量是以 O 为起点、以 A 为终点的向量 OA。若 A 的坐标为 (x, y)，其位置向量为 a = x i + y j。连接两点 A 和 B 的向量可表示为 AB = OB – OA。位置向量为坐标几何带来了强大的代数结构。
7. Unit Vectors and the Base Vectors i, j | 单位向量与基向量 i, j
A unit vector has magnitude 1. For any non‑zero vector v, the unit vector in its direction is v̂ = v / |v|. The standard base vectors are i = (1, 0) and j = (0, 1); every plane vector can be uniquely written as a = a₁ i + a₂ j. This basis simplifies calculations and is the foundation for vector components in later topics such as kinematics and linear transformations.
单位向量的模为 1。对于任何非零向量 v，沿其方向的单位向量为 v̂ = v / |v|。标准基向量是 i = (1, 0) 和 j = (0, 1)；每个平面向量都可以唯一地写为 a = a₁ i + a₂ j。这组基简化了运算，也是后续运动学和线性变换等内容中向量分量的基础。
8. The Dot Product (Scalar Product) | 点积（标量积）
The dot product of two vectors a = a₁ i + a₂ j and b = b₁ i + b₂ j is defined in two equivalent ways:
a · b = |a| |b| cos θ = a₁b₁ + a₂b₂
where θ is the angle between the vectors when placed tail‑to‑tail. The dot product is a scalar; it is positive for acute angles, zero for right angles, and negative for obtuse angles. It is widely used to test perpendicularity and to project one vector onto another.
两个向量 a = a₁ i + a₂ j 与 b = b₁ i + b₂ j 的点积有两种等价定义：
a · b = |a| |b| cos θ = a₁b₁ + a₂b₂
其中 θ 是两向量尾端相接时的夹角。点积是一个标量；锐角时为正，直角时为零，钝角时为负。它广泛用于检验垂直关系以及将一个向量投影到另一个向量上。
9. Angle Between Two Vectors | 两向量之间的夹角
Rearranging the dot product formula gives the cosine of the angle:
cos θ = (a · b) / (|a| |b|)
Two non‑zero vectors are perpendicular (orthogonal) if and only if a · b = 0. This condition appears frequently in IB exam questions, often requiring you to find unknown components or to show that lines are perpendicular.
将点积公式变形即得夹角余弦：
cos θ = (a · b) / (|a| |b|)
两个非零向量垂直（正交）当且仅当 a · b = 0。这一条件在 IB 试题中频繁出现，常需求解未知分量或证明直线垂直。
10. Vector Equation of a Straight Line | 直线的向量方程
A line in the plane can be described by a position vector a to a point on the line and a direction vector b that gives its slope. The vector equation is:
r = a + t b, t ∈ ℝ
Here r ether points lie on a line, and writing parametric equations x = a₁ + t b₁, y = a₂ + t b₂.
平面中的一条直线可用直线上某点的位置向量 a 以及给定斜率的方向向量 b 描述。其向量方程为：
r = a + t b, t ∈ ℝ
当参数 t 变化时，r 扫出直线上所有点。这一形式对于求交点、判断点是否在直线上以及写出参数方程 x = a₁ + t b₁, y = a₂ + t b₂ 极为实用。
11. Midpoints, Distances and Ratios | 中点、距离与比例分割
The midpoint M of points A and B with position vectors a and b has position vector m = ( a + b ) / 2. The distance between A and B is |AB| = |b – a|. Internal and external division of a segment by a ratio λ : μ can be expressed using section formulae in vector form, which are neat generalisations of the coordinate versions.
若点 A 与 B 的位置向量为 a 和 b，则中点 M 的位置向量为 m = ( a + b ) / 2。A 与 B 的距离为 |AB| = |b – a|。按比例 λ : μ 内分或外分线段可用向量形式的定比分点公式表示，这是坐标形式的优雅推广。
12. Common IB Problem–Solving Patterns | 常见 IB 解题模式
Typical IB questions ask you to prove that three points are collinear (check that one vector is a scalar multiple of another), determine the angle between two vectors, find the value of k that makes two vectors perpendicular, or derive the intersection of two lines given in vector form. A confident command of both geometric reasoning and algebraic manipulation is key. Always sketch a diagram, label your vectors clearly, and double‑check the quadrant when computing angles.
典型的 IB 问题要求证明三点共线（检验一个向量是否为另一向量的标量倍数）、确定两向量夹角、求使两向量垂直的 k 值，或推导两条以向量形式给出的直线的交点。同时掌握几何推理与代数运算至关重要。务必画出示意图，清晰标注向量，并在计算角度时检查象限。
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