Vectors and Vector Notation | 向量与向量记法

📚 Vectors and Vector Notation | 向量与向量记法

Vectors are quantities that have both magnitude and direction. In IGCSE Mathematics, vectors are used to describe translations, positions, and geometric relationships in two dimensions. This article covers the essential ideas of vectors and vector notation for Edexcel IGCSE students.

向量是既有大小又有方向的量。在 IGCSE 数学中，向量用于描述平面内的平移、位置以及几何关系。本篇文章围绕 Edexcel IGCSE 考纲，系统讲解向量的核心概念与标准记法。

1. What Is a Vector? | 什么是向量

A vector is a mathematical object with both magnitude (length) and direction. For example, a displacement of 5 km north is a vector, while a speed of 5 km/h is not a vector because it lacks direction.

向量是一种既有大小（长度）又有方向的数学对象。例如，“向北移动 5 公里”是一个向量，而“5 公里/小时”的速度不是向量，因为它没有方向。

In contrast, a scalar is a quantity that has only magnitude. Mass, temperature, and time are scalars. Vectors are usually represented by bold letters, underlined letters, or arrows above letters.

相比之下，标量是只有大小的量。质量、温度和时间都是标量。向量通常用粗体字母、下划线字母或字母上方带箭头来表示。

Bold: v | 粗体：v
Underlined: v̲ | 下划线：v̲
Arrow: v⃗ or AB⃗ | 箭头：v⃗ 或 AB⃗

2. Column Vectors | 列向量

In two dimensions, a vector can be written as a column vector. The top number is the horizontal component, and the bottom number is the vertical component.

在二维平面中，向量可以写成列向量的形式。上方的数表示水平分量，下方的数表示垂直分量。

( 3 )
( 2 )

This means 3 units to the right and 2 units up. A negative horizontal component means left, and a negative vertical component means down.

这表示向右 3 个单位、向上 2 个单位。水平分量为负表示向左，垂直分量为负表示向下。

For example, the column vector (−4, 1) written vertically means 4 units left and 1 unit up.

例如，列向量 (−4, 1) 表示向左 4 个单位、向上 1 个单位。

3. Vector Notation from Points | 由点表示的向量记法

If point A has coordinates (x₁, y₁) and point B has coordinates (x₂, y₂), then the vector AB⃗ is found by subtracting the coordinates of A from B.

若点 A 的坐标为 (x₁, y₁)，点 B 的坐标为 (x₂, y₂)，则向量 AB⃗ 由 B 的坐标减去 A 的坐标得到。

AB⃗ = ( x₂ − x₁ )
( y₂ − y₁ )

This vector represents the translation from A to B. The order matters: AB⃗ is not the same as BA⃗.

该向量表示从 A 到 B 的平移。顺序很重要：AB⃗ 和 BA⃗ 不同。

AB⃗ means from A to B | AB⃗ 表示从 A 到 B
BA⃗ means from B to A | BA⃗ 表示从 B 到 A
BA⃗ = − AB⃗ | BA⃗ = − AB⃗

4. Magnitude of a Vector | 向量的大小

The magnitude of a vector is its length. For a column vector (x, y), the magnitude is given by Pythagoras’ theorem.

向量的大小就是它的长度。对于列向量 (x, y)，其大小由勾股定理给出。

|v| = √( x² + y² )

For example, the magnitude of the vector (3, 4) is √(3² + 4²) = √25 = 5.

例如，向量 (3, 4) 的大小为 √(3² + 4²) = √25 = 5。

The magnitude is always positive or zero. It is never negative. The notation |v| or |AB⃗| is used for magnitude.

大小永远是非负数，不可能为负。记法 |v| 或 |AB⃗| 表示向量的大小。

5. Equal 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. In column form, equal vectors have exactly the same components.

两个向量相等当且仅当它们的大小和方向都相同。用列向量表示时，相等向量的各个分量完全相同。

Parallel vectors have the same direction but may have different magnitudes. A negative vector has the same magnitude but exactly the opposite direction.

平行向量方向相同但大小可以不同。负向量大小相同，但方向恰好相反。

If v = ( a, b ), then −v = ( −a, −b )

If v = (a b) written as a column, then −v = (−a −b) as a column. Geometrically, −v points in the opposite direction.

若 v = (a, b) 为列向量，则 −v = (−a, −b) 作为列向量。几何上，−v 指向相反方向。

6. Addition and Subtraction of Vectors | 向量的加法与减法

To add two vectors, add their corresponding components. To subtract, subtract the corresponding components.

两个向量相加时，只需将它们对应的分量相加；相减时，将对应分量相减。

(a, b) + (c, d) = (a+c, b+d)

(a, b) − (c, d) = (a−c, b−d)

Geometrically, adding vectors can be shown by the triangle law or the parallelogram law. If you travel along one vector and then another, the resultant is the sum.

在几何上，向量加法可用三角形法则或平行四边形法则表示。如果先沿一个向量运动，再沿另一个向量运动，则合向量就是它们的和。

For example, (2, 3) + (1, −4) = (3, −1). This is a single resultant vector.

例如，(2, 3) + (1, −4) = (3, −1)。这就是一个合向量。

7. Scalar Multiplication | 标量乘法

When a vector is multiplied by a scalar (a number), every component is multiplied by that number.

当向量乘以一个标量（数）时，向量的每个分量都乘以该数。

k(a, b) = (ka, kb)

If k is positive, the direction stays the same; if k is negative, the direction reverses. The magnitude is multiplied by |k|.

若 k 为正数，方向不变；若 k 为负数，方向反转。向量的大小变为原来的 |k| 倍。

For example, 3(2, −5) = (6, −15). Also, −2(1, 4) = (−2, −8). Scalar multiplication is also called scaling a vector.

例如，3(2, −5) = (6, −15)。又如，−2(1, 4) = (−2, −8)。标量乘法也叫做向量的缩放。

8. Position Vectors | 位置向量

A position vector is a vector that starts from the origin O and points to a point P. It is usually written as p or OP⃗.

位置向量是从原点 O 指向点 P 的向量，通常记作 p 或 OP⃗。

If point P has coordinates (x, y), then its position vector is p = (x, y) as a column vector.

若点 P 的坐标为 (x, y)，则其位置向量是列向量 p = (x, y)。

The vector AB⃗ can be expressed using position vectors: AB⃗ = b − a, where a and b are the position vectors of A and B.

向量 AB⃗ 可以用位置向量表示：AB⃗ = b − a，其中 a 和 b 分别是 A 和 B 的位置向量。

AB⃗ = OB⃗ − OA⃗

9. Parallel Vectors and Collinear Points | 平行向量与共线点

Two vectors are parallel if one is a scalar multiple of the other. For example, (2, 6) is parallel to (1, 3) because (2, 6) = 2(1, 3).

两个向量平行当且仅当其中一个向量是另一个向量的标量倍。例如，(2, 6) 与 (1, 3) 平行，因为 (2, 6) = 2(1, 3)。

Three points A, B, and C are collinear if AB⃗ is a scalar multiple of BC⃗ (or equivalently of AC⃗), and they share a common point.

三点 A、B、C 共线，如果 AB⃗ 是 BC⃗ 的标量倍（等价地，是 AC⃗ 的标量倍），并且它们有一个公共点。

To prove collinearity, show that the vectors are parallel and share a common point. For example, if AB⃗ = 2BC⃗, then A, B, C are collinear.

要证明三点共线，需要说明相关向量平行且它们有公共点。例如，若 AB⃗ = 2BC⃗，则 A、B、C 共线。

10. Solving Vector Geometry Problems | 向量几何问题的求解

In vector geometry, you often describe paths using vectors. For example, if M is the midpoint of AB, then the position vector of M is the average of the position vectors of A and B.

在向量几何中，经常用向量描述路径。例如，若 M 是 AB 的中点，则 M 的位置向量是 A 和 B 位置向量的平均值。

OM⃗ = ( a + b ) / 2

For a point dividing AB in the ratio m : n, the position vector is given by the section formula.

对于按比例 m : n 分割 AB 的点，其位置向量由定比分点公式给出。

OP⃗ = ( n a + m b ) / ( m + n )

This formula is very useful in exam problems involving ratios on straight lines.

这个公式在涉及直线上比例关系的考试题目中非常有用。

11. Common Exam Tips | 常见考试要点

Always write vectors clearly with correct notation, such as AB⃗ or column form. | 始终使用正确的记法书写向量，例如 AB⃗ 或列向量形式。
When subtracting vectors, be careful with signs. AB⃗ = b − a, not a − b. | 做向量减法时注意符号。AB⃗ = b − a，而不是 a − b。
Check whether a question asks for magnitude, direction, or components. | 仔细审题，确认题目要求的是大小、方向还是分量。
For parallel vectors, use scalar multiples. | 对于平行向量，使用标量倍关系。
Draw a diagram when possible to help visualise vector paths. | 尽可能画图来帮助理解向量路径。
In a ratio problem, use the section formula correctly. | 在比例问题中，正确使用定比分点公式。

12. Summary | 总结

Vectors are essential tools in IGCSE mathematics. You must understand column vectors, magnitude, addition, subtraction, scalar multiplication, position vectors, and how to work with parallel and collinear vectors.

向量是 IGCSE 数学的重要组成部分。你需要理解列向量、向量大小、加法、减法、标量乘法、位置向量，以及平行和共线向量的处理方法。

Practise transforming between geometric descriptions and column vectors, and always show clear working in exams.

多加练习在几何描述与列向量之间进行转换，并在考试中展示清晰的解题过程。

Mastering vector notation now will prepare you for more advanced vector topics in A-Level and beyond.

现在掌握好向量记法，将为你以后学习 A-Level 及更高级的向量内容打下坚实基础。

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Vectors and Vector Notation | 向量与向量记法