📚 Elementary Operations with Vectors | 向量的基本运算
Vectors are fundamental in IB Mathematics, appearing in topics from coordinate geometry to mechanics and linear algebra. Understanding elementary operations with vectors – addition, subtraction, scalar multiplication, dot product, and more – builds a solid foundation for solving problems involving forces, velocities, displacements, and geometric relationships. This article walks through each key operation, providing clear definitions, worked examples, and links to IB-style reasoning.
向量是IB数学中的基础，出现在从坐标几何到力学和线性代数的诸多课题中。掌握向量的基本运算——加法、减法、标量乘法、点积等——为解决涉及力、速度、位移和几何关系的问题奠定了扎实的基础。本文逐一介绍每个关键运算，提供清晰的定义、计算实例，并与IB风格的推理相联系。
1. Definition and Notation of Vectors | 向量的定义与记号
A vector is a quantity that possesses both magnitude (size) and direction. Examples include displacement, velocity, and force. In contrast, a scalar has only magnitude, such as mass or temperature.
向量是既有大小又有方向的量，例如位移、速度和力。相反，标量只有大小，如质量或温度。
Vectors are commonly denoted by boldface letters – for instance, v – or by an arrow above the letter. In two dimensions, a vector can be expressed in component form as v = (v₁, v₂) or using the standard unit vectors i and j: v = v₁i + v₂j. In three dimensions, we add the unit vector k: v = v₁i + v₂j + v₃k.
向量通常用粗体字母表示，例如v，或在字母上方加箭头。在二维中，向量可以用分量形式表示为v = (v₁, v₂) 或使用标准单位向量 i 和 j：v = v₁i + v₂j。在三维中，我们加入单位向量 k：v = v₁i + v₂j + v₃k。
Two vectors are equal if and only if their corresponding components are equal; that is, they have the same magnitude and direction, regardless of their initial position.
当且仅当两个向量的对应分量相等时，它们相等；即它们具有相同的大小和方向，无论起点在何处。
2. Vector Addition | 向量加法
Vector addition follows the triangle law or the parallelogram law. To add vectors a and b graphically using the triangle law, place the tail of b at the head of a. The sum a + b is the vector drawn from the tail of a to the head of b.
向量加法遵循三角形法则或平行四边形法则。用三角形法则作图相加时，将b的起点与a的终点重合，和向量a + b 就是从a的起点指向b的终点的向量。
Alternatively, with the parallelogram law, place the tails of a and b together, complete the parallelogram, and the diagonal emerging from the common tail represents a + b.
或者，使用平行四边形法则：将a和b的起点重合，作平行四边形，从公共起点引出的对角线即为a + b。
In component form, if a = (a₁, a₂) and b = (b₁, b₂), then the sum is obtained by adding corresponding components:
a + b = (a₁ + b₁, a₂ + b₂)
在分量形式中，若a = (a₁, a₂)，b = (b₁, b₂)，则求和只需将对应分量相加。
3. Vector Subtraction | 向量减法
Subtracting vector b from a is equivalent to adding the negative of b: a - b = a + (-b). The vector -b has the same magnitude as b but points in the opposite direction.
从a减去b等同于加上b的相反向量：a - b = a + (-b)。向量 -b 与 b 大小相等但方向相反。
Using components, subtraction is straightforward:
a - b = (a₁ - b₁, a₂ - b₂)
使用分量，减法简单直接：对应分量相减。
Geometrically, the difference a - b can be represented as the other diagonal of the parallelogram formed by a and b, directed from the head of b to the head of a when tails coincide.
从几何上看，差a - b 可表示为a和b所构成的平行四边形的另一条对角线，当起点重合时，其方向从b的终点指向a的终点。
4. Scalar Multiplication | 标量乘法
When a vector v is multiplied by a scalar k, the result kv is a vector whose magnitude is scaled by the factor |k|. If k > 0, the direction remains unchanged; if k < 0, the direction is reversed.
当向量v乘以标量 k 时，结果 kv 是一个大小按因子 |k| 缩放的新向量。若 k > 0，方向不变；若 k < 0，方向相反。
In component terms, each component is multiplied by k:
k v = (k v₁, k v₂, k v₃)
在分量形式中，每个分量都乘以 k。
Scalar multiplication is distributive over vector addition: k(a + b) = ka + kb. This property is extremely useful in solving vector equations and simplifying expressions.
标量乘法对向量加法满足分配律：k(a + b) = ka + kb。这一性质在求解向量方程和化简表达式时非常有用。
5. Magnitude of a Vector | 向量的大小（模）
The magnitude (or length) of a vector v = (x, y) in two dimensions is given by the distance formula:
|v| = √x² + y²
二维向量v = (x, y) 的模（长度）由距离公式给出。
In three dimensions, for v = (x, y, z), the magnitude is:
|v| = √x² + y² + z²
三维向量v = (x, y, z) 的模为 √x² + y² + z²。
The magnitude of kv is |k|·|v|. Moreover, the triangle inequality states that |a + b| ≤ |a| + |b|, with equality only when a and b point in the same direction.
kv 的模等于 |k|·|v|。此外，三角不等式指出 |a + b| ≤ |a| + |b|，仅当 a 和 b 同向时取等号。
6. Unit Vectors | 单位向量
A unit vector is a vector whose magnitude is exactly 1. Unit vectors describe direction without scaling effects. Any non‑zero vector v can be converted into a unit vector û (often written as û) by dividing by its own magnitude:
û = v / |v|
单位向量是模恰好为1的向量。它描述方向，而不受大小影响。任何非零向量v都可除以自身模长化为单位向量。
The standard basis vectors i, j (and k in 3D) are unit vectors pointing in the positive coordinate directions. For instance, i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1).
标准基向量 i、j（三维中还有 k）是指向坐标轴正方向的单位向量。例如 i = (1, 0, 0)，j = (0, 1, 0)，k = (0, 0, 1)。
Using a unit vector, any vector can be expressed as the product of its magnitude and a direction vector: v = |v| û. This decomposition is valuable in physics when separating size from orientation.
利用单位向量，任何向量都可表示为其模与方向向量的乘积：v = |v| û
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