📚 Operations with Plane Vectors | 平面向量的运算
In IB Mathematics, plane vectors are essential for describing translations, forces, velocities and many geometric relationships. Mastering operations such as addition, scalar multiplication, dot product and finding magnitudes allows you to solve problems in both pure and applied contexts. This article covers all key operations with vectors in two dimensions, with clear explanations and examples.
在IB数学中，平面向量是描述平移、力、速度以及许多几何关系的基本工具。掌握加法、标量乘法、点积以及求模等运算是解决纯数学和应用问题的基础。本文涵盖了二维向量的所有关键运算，并附有清晰的解释和示例。
1. Introduction to Plane Vectors | 平面向量简介
A plane vector is a quantity that possesses both magnitude and direction. Geometrically, it can be drawn as a directed line segment from an initial point to a terminal point. The length of the segment represents the magnitude, and the arrow indicates the direction. In component form, a vector v is written as v nd y are the horizontal and vertical components relative to a Cartesian coordinate system.
平面向量是一个既有大小又有方向的量。几何上，它可以用从起点到终点的有向线段表示，线段长度代表大小，箭头指示方向。用分量形式表示时，向量 v 写作 v = (x, y)，其中 x 和 y 分别是相对于笛卡尔坐标系的水平和垂直分量。
Two vectors are equal if they have the same length and the same direction, regardless of where they are positioned. The zero vector, denoted by 0, has magnitude 0 and no specific direction; it acts as the additive identity. The negative of a vector v is -v, which has the same magnitude but opposite direction.
两个向量如果长度相同且方向相同则相等，无论它们处于什么位置。零向量记作 0，大小为0且没有特定方向，它是加法的单位元。向量 v 的负向量是 -v，大小相等但方向相反。
2. Vector Addition and Subtraction | 向量的加法与减法
Vector addition can be performed geometrically using the triangle or parallelogram law, or algebraically by adding corresponding components. For two vectors a = (a₁, a₂) and b = (b₁, b₂), their sum is a + b = (a₁ + b₁, a₂ + b₂). The operation is commutative and associative.
向量加法可以用三角形法则或平行四边形法则几何地完成，也可以通过对分量代数相加来实现。对于两个向量 a = (a₁, a₂) 和 b = (b₁, b₂)，它们的和为 a + b = (a₁ + b₁, a₂ + b₂)。此运算满足交换律和结合律。
a + b = (a₁ + b₁, a₂ + b₂)
Vector subtraction is defined as adding the negative: a - b = a + (-b). In component terms, a - b = (a₁ - b₁, a₂ - b₂). Geometrically, the difference a - b is the vector from the tip of b to the tip of a when placed tail-to-tail.
向量减法定义为加上负向量：a - b = a + (-b)。用分量表示为 a - b = (a₁ - b₁, a₂ - b₂)。几何上，当两个向量尾对尾放置时，差向量 a - b 是从 b 的末端指向 a 末端的向量。
3. Scalar Multiplication | 标量乘法
When a vector v = (x, y) is multiplied by a scalar k, the result is a vector kv = (kx, ky). The magnitude is scaled by |k| k = 0, the result is the zero vector. This operation distributes over vector addition: k(a + b) = ka + kb.
当向量 v = (x, y) 乘以标量 k 时，结果向量为 kv = (kx, ky)。模长按 |k| 缩放，若 k > 0 方向不变，若 k < 0 则方向反向。k = 0 时结果为零向量。此运算对向量加法满足分配律：k(a + b) = ka + kb。
Scalar multiplication is fundamental for describing parallel vectors. Two non-zero vectors are parallel if one is a scalar multiple of the other, i.e., a = λb for some non-zero λ ∈ ℝ.
标量乘法是描述平行向量的基础。两个非零向量平行当且仅当一个是另一个的标量倍数，即存在非零实数 λ 使得 a = λb。
4. Magnitude of a Vector | 向量的模
The magnitude (or length) of a vector v = (x, y) is denoted by |v| or ||v|| and is calculated using the Pythagorean theorem: |v| = √x² + y². This value is always non-negative. The magnitude represents the distance from the initial point to the terminal point. For a vector AB from A(x₁, y₁) to B(x₂, y₂), its components are (x₂ - x₁, y₂ - y₁) and its magnitude equals the distance between A and B.
向量 v = (x, y) 的模（长度）记作 |v| 或 ||v||，可利用勾股定理计算：|v| = √x² + y²。该值始终非负。模代表从起点到终点的距离。对于从 A(x₁, y₁) 到 B(x₂, y₂) 的向量 AB，其分量为 (x₂ - x₁, y₂ - y₁)，其模等于 A 与 B 之间的距离。
|v| = √x² + y²
A vector with magnitude 1 is called a unit vector. Any non-zero vector can be normalised to a unit vector in the same direction by dividing by its magnitude.
模长为1的向量称为单位向量。任何非零向量可通过除以其模长化为同方向的单位向量。
5. Unit Vectors and Direction | 单位向量与方向
Given a non-zero vector v, the unit vector in the direction of v is û = v / |v|. This vector has magnitude 1. In two dimensions, the standard basis vectors are i = (1, 0) and j = (0, 1). Any vector v = (x, y) can be expressed as xi + yj.
给定非零向量 v，沿 v 方向的单位向量为 û = v / |v|。该向量的模为1。在二维空间中，标准基向量为 i = (1, 0) 和 j = (0, 1)。任何向量 v = (x, y) 都可以表示为 xi + yj。
The direction of a vector can also be described by the angle θ it makes with the positive x-axis, measured counterclockwise. If v θ = y/x (with appropriate quadrant consideration). The components can be recovered as x = |v| cos θ and y = |v| sin θ.
向量的方向也可用它与 x 轴正方向的夹角 θ（逆时针测量）来描述。若 v = (x, y)，则 tan θ = y/x（需考虑象限）。分量可通过 x = |v| cos θ、y = |v| sin θ 反求。
6. Position Vectors | 位置向量
A position vector is a vector that starts at the origin O(0,0) and ends at a point P(x, y). It is denoted by OP or simply p onvenient way to describe points and are fundamental in vector geometry. The vector from point A to point B is given by AB = b - a, where a and b are the position vectors of A and B.
位置向量是以原点 O(0,0) 为起点、以点 P(x, y) 为终点的向量，记作 OP 或简写为 p = (x, y)。位置向量为描述点提供了便利，是向量几何的基础。从点 A 到点 B 的向量为 AB = b - a，其中 a 和 b 分别是 A 和 B 的位置向量。
Using position vectors, the midpoint M of segment AB has position vector (a + b)/2. Collinearity of points A, B and C can be checked by verifying that AB and AC are parallel, i.e., AB = k AC for some scalar k.
利用位置向量，线段 AB 的中点 M 的位置向量为 (a + b)/2。通过验证 AB 与 AC 平行（即 AB = k AC）可判断 A、B、C 三点共线。
7. Dot Product (Scalar Product) | 数量积（点积）
The dot product of two vectors a = (a₁, a₂) and b = (b₁, b₂) is defined as a·b = a₁b₁ + a₂b₂, which is a scalar. Equivalently, a·b = |a||b| cos θ, where θ is the angle between the vectors. This second form is extremely useful for finding angles and testing orthogonality.
两个向量 a = (a₁, a₂) 和 b = (b₁, b₂) 的点积定义为 a·b = a₁b₁ + a₂b₂，结果是一个标量。等价地，a·b = |a||b| cos θ，其中 θ 是两向量间的夹角。第二种形式对于求夹角和判断垂直性极为有用。
a·b = a₁b₁ + a₂b₂ = |a||b| cos θ
The dot product is commutative, distributive over addition, and k(a·b) = (ka)·b = a·(kb). Two non-zero vectors are perpendicular (orthogonal) if and only if their dot product is zero.
点积满足交换律、对加法的分配律，且 k(a·b) = (ka)·b = a·(kb)。两个非零向量垂直（正交）当且仅当它们的点积为零。
8. Angle Between Two Vectors | 两向量间的夹角
By rearranging the geometric form of the dot product, the angle θ between two non-zero vectors a and b is given by cos θ = (a·b) / (|a||b| ame direction; cos θ = -1 means they are opposite; cos θ = 0 means they are perpendicular.
通过整理点积的几何形式，两个非零向量 a 和 b 之间的夹角 θ 由 cos θ = (a·b) / (|a||b|) 给出。该公式给出的角度总是在 [0, π]（或 0° 到 180°）范围内。夹角衡量了两个向量的对齐程度：cos θ = 1 表示同向平行；cos θ = -1 表示反向；cos θ = 0 表示垂直。
cos θ = (a·b) / (|a||b|)
In component form, this becomes cos θ = (a₁b₁ + a₂b₂) / (√a₁² + a₂² √b₁² + b₂²). This formula allows you to compute the angle directly from coordinates without sketching.
用分量形式表示即为 cos θ = (a₁b₁ + a₂b₂) / (√a₁² + a₂² √b₁² + b₂²)。利用该公式可直接从坐标计算夹角，无需绘图。
9. Orthogonal and Parallel Vectors | 垂直与平行向量
Two vectors are perpendicular (orthogonal) if and only if a·b = 0. This is a clean algebraic test that does not require finding the angle. For example, a = (3, 4) and b = (-4, 3) are orthogonal because 3·(-4) + 4·3 = 0.
两个向量垂直（正交）当且仅当 a·b = 0。这是一个不需要求角度的简洁代数检验法。例如，a = (3, 4) 与 b = (-4, 3) 正交，因为 3·(-4) + 4·3 = 0。
Two non-zero vectors are parallel if one is a scalar multiple of the other: a = kb for some k ∈ ℝ a₁b₂ - a₂b₁ = 0. Parallel vectors have an angle of 0 or π, and their dot product satisfies |a·b| = |a||b|.
两个非零向量平行当一个向量是另一个的标量倍数：存在实数 k 使得 a = kb。这也意味着二维情形中的类比叉乘（行列式）为零：a₁b₂ - a₂b₁ = 0。平行向量的夹角为 0 或 π，其点积满足 |a·b| = |a||b|。
10. Applications in Geometry | 几何中的应用
Vector operations are powerful for solving geometric problems. The midpoint of AB is given by (a + b)/2. The centroid of triangle with vertices at A, B, C is (a + b + c)/3. Using dot products, one can find the foot of perpendiculars, projections, and angles between lines.
向量运算在解决几何问题中功能强大。线段 AB 的中点为 (a + b)/2。以 A、B、C 为顶点的三角形重心为 (a + b + c)/3。借助点积可以求垂足、投影以及直线间夹角。
The scalar projection of a onto b is (a·b) / |b|, and the vector projection is ((a·b) / |b|²) b. These are used to decompose a vector into parallel and perpendicular components relative to another vector.
a 在 b 上的标量投影为 (a·b) / |b|，向量投影为 ((a·b) / |b|²) b。这些可将一个向量分解为相对于另一向量的平行分量和垂直分量。
Vectors also allow elegant proofs of geometric theorems, such as the diagonals of a parallelogram bisecting each other or the concurrency of medians in a triangle, without relying on coordinate geometry.
向量还可使几何定理的证明变得简洁，如平行四边形的对角线互相平分或三角形中线共点，而无需依赖坐标几何。
11. Summary and Key Formulas | 总结与关键公式
The following table summarises the main operations with plane vectors. Mastering these will give you confidence in tackling IB exam questions on vectors.
下表总结了平面向量的主要运算。熟练掌握这些运算将使你在应对IB向量考试题目时充满信心。
| Operation | Formula / Description |
| Addition | a + b = (a₁ + b₁, a₂ + b₂) |
| Subtraction | a - b = (a₁ - b₁, a₂ - b₂) |
| Scalar Multiplication | kv = (kx, ky) |
| Magnitude | |v| = √x² + y² |
| Unit Vector | û = v / |v| |
| Position Vector of AB | AB = b - a |
| Dot Product | a·b = a₁b₁ + a₂b₂ = |a||b| cos θ |
| Angle Between Vectors | cos θ = (a·b) / (|a||b|) |
| Orthogonality Test | a·b = 0 ⇔ perpendicular |
| Parallelism Test | a = kb or a₁b₂ - a₂b₁ = 0 |
Remember to always consider the geometric interpretation alongside the algebraic manipulation. This dual understanding is key to success in IB Mathematics.
请始终同时考虑几何解释和代数运算。这种双重理解是IB数学取得成功的关键。
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