📚 8A Introduction to Vectors | 8A 向量入门
Vectors are a fundamental concept in mathematics and physics, essential for describing quantities that have both magnitude and direction. In IB Mathematics, mastering vectors from the ground up is crucial for topics in geometry, mechanics, and beyond. This article provides a thorough introduction to vectors, covering notation, basic operations, and important properties.
向量是数学和物理学中的一个基本概念，对于描述既有大小又有方向的量至关重要。在IB数学中，从基础掌握向量对于几何、力学及其他专题尤为重要。本文全面介绍向量，涵盖符号、基本运算和重要性质。
1. What is a Vector? | 什么是向量？
A vector is a quantity that has both magnitude (size) and direction. In contrast, a scalar has only magnitude. For example, displacement, velocity, and force are vector quantities; distance, speed, and mass are scalars.
向量是具有大小（模）和方向的量。相反，标量只有大小。例如，位移、速度和力是向量；距离、速率和质量是标量。
In diagrams, vectors are represented by arrows. The length of the arrow represents the magnitude, and the arrowhead points in the direction of the vector.
在图中，向量用箭头表示。箭头的长度表示模，箭头指向向量的方向。
2. Notation and Representation | 符号与表示
Vectors can be written in several ways. A vector from point A to point B is written as AB→ or simply as a bold lowercase letter such as v. In handwriting, we often use a wavy underline or an arrow above: v~ or v→.
向量可以有多种写法。从点A到点B的向量写作AB→，或简写为粗体小写字母如v。手写时，常在字母下方加波浪线或上方加箭头：v~ 或 v→。
In typed material, boldface is standard: a, b, c. For magnitude, we use |v| or ||v||.
在印刷体中，粗体是标准形式：a, b, c。向量的模用 |v| 或 ||v|| 表示。
3. Magnitude of a Vector | 向量的模
The magnitude of a vector v is its length, denoted |v|. For a vector in component form v = (x, y) in 2D, the magnitude is |v| = √x² + y². In 3D with v = (x, y, z), |v| = √x² + y² + z².
向量v的模是其长度，记作 |v|。对于二维分量形式 v = (x, y)，模为 |v| = √x² + y²。对于三维 v = (x, y, z)，| v | = √x² + y² + z²。
The magnitude is always non-negative and represents the 'size' of the vector regardless of direction.
模永远非负，表示向量的“大小”而不考虑方向。
4. Zero Vector and Unit Vectors | 零向量与单位向量
The zero vector, denoted 0, has magnitude 0 and no defined direction. It is the additive identity: v + 0 = v.
零向量记作0，模为0，方向无定义。它是加法单位元：v + 0 = v。
A unit vector has a magnitude of 1. Any non-zero vector v can be converted into a unit vector â in the same direction by dividing by its magnitude: â = v / |v|.
单位向量的模为1。任何非零向量v除以它的模，可得到同方向的单位向量 â：â = v / |v|。
The standard unit vectors in 2D are i = (1, 0) and j = (0, 1). In 3D, we add k = (0, 0, 1).
二维标准单位向量是 i = (1, 0) 和 j = (0, 1)。三维则加上 k = (0, 0, 1)。
5. Position Vectors | 位置向量
The position vector of a point P relative to the origin O is the vector OP on of P in terms of displacement from the origin. For point (x, y), the position vector is p = xi + yj or simply (x, y).
一点 P 相对于原点 O 的位置向量是OP→，它用从原点的位移给出 P 的位置。对于点 (x, y)，位置向量是 p = xi + yj，或简写为 (x, y)。
Position vectors are fundamental for expressing other vectors between points: AB→ = OB→ - OA→.
位置向量是表示两点间向量的基础：AB→ = OB→ - OA→。
6. Equal Vectors and Negative Vectors | 相等向量与相反向量
Two vectors are equal if they have the same magnitude and the same direction. Their starting points do not matter; only the displacement matters.
两个向量相等当且仅当它们模相等且方向相同。起点无关紧要，只有位移重要。
The negative of a vector v is denoted -v. It has the same magnitude as v but points in the opposite direction. So v + (-v) = 0.
向量 v 的相反向量记作 -v，它与 v 模相等但方向相反。因此 v + (-v) = 0。
7. Scalar Multiplication | 标量乘法
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