📚 Basic Concepts of Vectors in a Plane | 平面向量的基本概念
Vectors are one of the most fundamental tools in IB Mathematics, providing a bridge between geometry and algebra. In this article, we will explore the essential definitions and operations that form the foundation of vector geometry in the plane.
向量是IB数学中最基础的工具之一，它连接了几何与代数。在本文中，我们将探索平面向量几何中构成基础的核心定义与运算。
1. What Is a Vector? | 什么是向量？
A vector is a quantity that has both magnitude (size) and direction. For example, displacement, velocity, and force are all vector quantities.
向量是既有大小又有方向的量。例如，位移、速度和力都是向量。
In contrast, a scalar only has magnitude, such as temperature, mass, or speed. A vector is usually represented by a directed line segment, with an arrow indicating its direction.
相比之下，标量只有大小，例如温度、质量或速率。向量通常用有向线段表示，箭头指示方向。
overrightarrowAB denotes the vector from point A to point B.
overrightarrowAB 表示从点 A 到点 B 的向量。
v⃗ = (x, y)ᵀ in component form.
v⃗ = (x, y)ᵀ 为分量形式。
2. Vector vs Scalar | 向量与标量的区别
A scalar is described entirely by a single real number, while a vector requires both a number and a direction.
标量可以用一个实数完全描述，而向量则需要数值和方向两个要素。
- Scalar examples: mass 5 kg, temperature 20 °C, speed 60 km/h.
- 向量示例：质量 5 kg、温度 20 °C、速率 60 km/h。
- Vector examples: displacement 5 km north, velocity 60 km/h east, force 10 N downward.
- 向量示例：位移 5 km 向北、速度 60 km/h 向东、力 10 N 向下。
3. Equal Vectors and Negative Vectors | 相等向量与负向量
Two vectors are equal if they have the same magnitude and the same direction, regardless of their starting point.
两个向量相等当且仅当它们的大小相同且方向相同，与起点无关。
The negative of a vector v⃗, written as -v⃗, has the same magnitude but exactly the opposite direction.
向量 v⃗ 的负向量记作 -v⃗，它的大小相同，但方向完全相反。
If AB⃗ = CD⃗, then the two directed segments are parallel, equal in length, and point in the same direction.
若 AB⃗ = CD⃗，则两条有向线段平行、等长且同向。
4. Magnitude of a Vector | 向量的模
The magnitude (or length) of a vector v⃗ = (x, y)ᵀ is given by the distance formula.
向量 v⃗ = (x, y)ᵀ 的模（即长度）由距离公式给出。
|v⃗| = √x2 + y2
|v⃗| = √x2 + y2
For example, if v⃗ = (3, -4)ᵀ, then |v⃗| = √32 + (-4)2 = 5.
例如，若 v⃗ = (3, -4)ᵀ，则 |v⃗| = √32 + (-4)2 = 5。
5. Unit Vectors | 单位向量
A unit vector is a vector with magnitude exactly 1. It is often used to indicate direction only.
单位向量是模等于 1 的向量，通常仅用于表示方向。
To find a unit vector in the direction of v⃗, divide v⃗ by its magnitude:
求与 v⃗ 同方向的单位向量，可将 v⃗ 除以它的模：
v̂ = v⃗|v⃗|
v̂ = v⃗|v⃗|
If v⃗ = (3, 4)ᵀ, then |v⃗| = 5, so v̂ = (35, 45)ᵀ.
若 v⃗ = (3, 4)ᵀ，则 |v⃗| = 5，因此 v̂ = (35, 45)ᵀ。
6. Position Vectors and Free Vectors | 位置向量与自由向量
A position vector is a vector that starts from the origin O. It describes the location of a point relative to O.
位置向量是以原点 O 为起点的向量，它描述了点相对于原点 O 的位置。
For a point A(x, y), its position vector is written as OA⃗ = (x, y)ᵀ.
对于点 A(x, y)，其位置向量记为 OA⃗ = (x, y)ᵀ。
A free vector can be translated anywhere in the plane without changing its value, because only its magnitude and direction matter.
自由向量可以在平面内任意平移而不改变其值，因为只有大小和方向起作用。
7. Vector Addition | 向量的加法
Vector addition can be performed using algebra, the triangle rule, or the parallelogram rule.
向量加法可以用代数法、三角形法则或平行四边形法则进行。
Algebraically, if a⃗ = (a1, a2)ᵀ and b⃗ = (b1, b2)ᵀ, then
代数上，若 a⃗ = (a1, a2)ᵀ，b⃗ = (b1, b2)ᵀ，则
a⃗ + b⃗ = (a1 + b1, a2 + b2)ᵀ
a⃗ + b⃗ = (a1 + b1, a2 + b2)ᵀ
Triangle rule: place the tail of b⃗ at the head of a⃗; the sum joins the tail of a⃗ to the head of b⃗.
三角形法则：将 b⃗ 的起点放在 a⃗ 的终点，和向量从 a⃗ 的起点指向 b⃗ 的终点。
Parallelogram rule: when both vectors start from the same point, the diagonal of the parallelogram gives the sum.
平行四边形法则：两个向量从同一起点出发，平行四边形的对角线即为和向量。
8. Scalar Multiplication | 数乘向量
Multiplying a vector by a scalar k changes its length but not its direction (unless k < 0, which reverses the direction).
向量乘以标量 k 会改变其长度但不改变方向（若 k < 0，则方向反转）。
If v⃗ = (x, y)ᵀ, then
若 v⃗ = (x, y)ᵀ，则
kv⃗ = (kx, ky)ᵀ
kv⃗ = (kx, ky)ᵀ
Also, |kv⃗| = |k| |v⃗|.
同时，|kv⃗| = |k| |v⃗|。
9. Linear Combinations and Basis Vectors | 线性组合与基本向量
In two dimensions, any vector can be expressed as a linear combination of two independent basis vectors. The standard basis vectors are
在二维平面中，任何向量都可以表示为两个不共线基向量的线性组合。标准基向量为
i⃗ = (1, 0)ᵀ, j⃗ = (0, 1)ᵀ
i⃗ = (1, 0)ᵀ, j⃗ = (0, 1)ᵀ
Thus any vector v⃗ = (x, y)ᵀ can be written as v⃗ = xi⃗ + yj⃗.
因此任何向量 v⃗ = (x, y)ᵀ 都可写成 v⃗ = xi⃗ + yj⃗。
This representation is extremely useful in solving problems involving geometry and motion.
这种表示在解决几何和运动问题时极为有用。
10. Parallel Vectors and Collinearity | 平行向量与共线
Two nonzero vectors are parallel if one is a scalar multiple of the other. That is, a⃗ = kb⃗ for some scalar k.
两个非零向量平行当且仅当其中一个可以写成另一个的标量倍，即存在标量 k 使 a⃗ = kb⃗。
If a⃗ = (a1, a2)ᵀ and b⃗ = (b1, b2)ᵀ are parallel, then
若 a⃗ = (a1, a2)ᵀ 和 b⃗ = (b1, b2)ᵀ 平行，则
a1⁄b1 = a2⁄b2 (provided denominators are nonzero)
a1⁄b1 = a2⁄b2（分母不为零时）
Collinearity: three points A, B, C are collinear if AB⃗ and AC⃗ are parallel, i.e. AB⃗ = tAC⃗ for some scalar t.
共线：三点 A、B、C 共线当且仅当 AB⃗ 与 AC⃗ 平行，即存在标量 t 使 AB⃗ = tAC⃗。
11. The Dot Product (Scalar Product) | 向量的点积（数量积）
The dot product of two vectors gives a scalar and is a key tool for finding angles between vectors.
两个向量的点积得到一个标量，它是求向量夹角的重要工具。
Algebraically, if a⃗ = (a1, a2)ᵀ and b⃗ = (b1, b2)ᵀ, then
代数上，若 a⃗ = (a1, a2)ᵀ，b⃗ = (b1, b2)ᵀ，则
a⃗ · b⃗ = a1 b1 + a2 b2
a⃗ · b⃗ = a1 b1 + a2 b2
Geometrically, a⃗ · b⃗ = |a⃗| |b⃗| cos θ, where θ is the angle between the vectors.
几何上，a⃗ · b⃗ = |a⃗| |b⃗| cos θ，其中 θ 是两个向量的夹角。
If two vectors are perpendicular, their dot product is zero: a⃗ · b⃗ = 0.
若两个向量垂直，则它们的点积为零：a⃗ · b⃗ = 0。
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