Vectors for Edexcel IGCSE Maths | 向量：爱德思 IGCSE 数学核心考点

📚 Vectors for Edexcel IGCSE Maths | 向量：爱德思 IGCSE 数学核心考点

Vectors are one of the most important topics in the Edexcel IGCSE Mathematics syllabus. They appear in Paper 1 and Paper 2, both as short questions and as challenging multi-part problems. This guide covers everything you need to know: notation, addition, subtraction, scalar multiplication, magnitude, position vectors, parallel vectors and ratio division problems.

向量是爱德思 IGCSE 数学大纲中最重要的话题之一。它在 Paper 1 和 Paper 2 中都会出现,既有小题,也有较有挑战性的多问大题。本指南涵盖了你需要掌握的全部内容:向量的表示、加法、减法、数乘、模长、位置向量、平行向量以及按比例分割线段问题。

1. What is a Vector? | 什么是向量？

A vector is a quantity that has both magnitude (size) and direction. For example, ‘walk 5 km east’ is a vector description, while ‘walk 5 km’ is only a scalar because it has no direction. Common vector quantities in mathematics include displacement, velocity and force.

向量是同时具有大小（模）和方向的量。例如,”向东走 5 公里”是向量描述,而”走 5 公里”只是标量,因为它没有方向。数学中常见的向量包括位移、速度和力。

A scalar is an ordinary number such as 5, −3 or 0.5. Multiplying a vector by a scalar changes its length but not its direction, unless the scalar is negative (then the direction is reversed).

标量就是普通的数,如 5、−3 或 0.5。用标量去乘向量,只会改变向量的长度,不会改变方向;除非标量为负（此时方向反转）。

In the Edexcel IGCSE syllabus, you will most often work with vectors in two dimensions, written algebraically or shown on a coordinate grid.

在爱德思 IGCSE 大纲中,最常处理的是二维向量,既可以用代数形式书写,也可以在坐标网格中画出来。

2. Vector Notation | 向量的表示方法

A column vector is written with the horizontal component on top and the vertical component on the bottom. In this guide we write a column vector as (x, y), meaning x units to the right and y units up. For example, (3, 2) means 3 units across and 2 units up; (−4, 1) means 4 units left and 1 unit up.

列向量书写时,水平分量在上、垂直分量在下。在本指南中,我们用 (x, y) 表示列向量,意思是从起点向右移动 x 个单位、向上移动 y 个单位。例如,(3, 2) 表示向右 3 个单位、向上 2 个单位;(−4, 1) 表示向左 4 个单位、向上 1 个单位。

A vector can be named using a bold lowercase letter such as a, or using two capital letters with a vector sign above, such as AB. The vector AB means ‘from A to B’. The vector BA is exactly opposite in direction and has the same magnitude, so AB = −BA.

向量可以用粗体小写字母命名,例如 a,也可以用两个大写字母并在上方加箭头,例如 AB。向量 AB 表示”从 A 到 B”。向量 BA 方向正好相反,大小相同,因此 AB = −BA。

The unit vectors i and j point in the positive x-direction and positive y-direction. Any vector (x, y) can be written as xi + yj. For example, (3, −2) = 3i − 2j.

单位向量 i 和 j 分别指向 x 轴正方向和 y 轴正方向。任何向量 (x, y) 都可以写成 xi + yj。例如,(3, −2) = 3i − 2j。

3. Vector Addition and Subtraction | 向量的加法与减法

To add two vectors, add the x-components together and the y-components together. For example, if a = (3, 1) and b = (2, 5), then a + b = (3 + 2, 1 + 5) = (5, 6). Geometrically, addition uses the ‘tip-to-tail’ rule: place the tail of b at the tip of a; the sum is the vector from the tail of a to the tip of b.

两个向量相加时,把 x 分量相加、y 分量相加。例如,若 a = (3, 1),b = (2, 5),则 a + b = (3 + 2, 1 + 5) = (5, 6)。几何上,加法采用”首尾相接”法则:把 b 的起点放在 a 的终点,和向量就是从 a 的起点指向 b 的终点。

Subtraction is similar: subtract the corresponding components. The vector from A to B is found using position vectors: AB = b − a, where a and b are the position vectors of A and B. This rule is used constantly in IGCSE vector questions.

减法类似:把对应的分量相减。从 A 到 B 的向量可以用位置向量求出:AB = b − a,其中 a 和 b 是 A 和 B 的位置向量。这个法则在 IGCSE 向量题中被反复使用。

A common mistake is to write a − b instead of b − a. Remember: the vector AB means ‘the journey from A to B’, which is ‘B minus A’. Drawing a small sketch with an arrow can help you avoid sign errors.

常见的错误是把 b − a 写成 a − b。记住:向量 AB 表示”从 A 到 B 的路径”,也就是”B 减 A”。画一个带箭头的小草图可以帮助你避免符号错误。

4. Scalar Multiplication | 向量的数乘

Multiplying a vector by a scalar k multiplies every component by k. If a = (x, y), then ka = (kx, ky). For example, 3 × (2, −1) = (6, −3).

向量乘以标量 k,相当于每个分量都乘以 k。若 a = (x, y),则 ka = (kx, ky)。例如,3 × (2, −1) = (6, −3)。

If k > 0, the direction is unchanged; if k < 0, the direction is reversed. The vector −a has the same length as a but points in the opposite direction. Two vectors are parallel if one is a scalar multiple of the other.

如果 k > 0,方向不变;如果 k < 0,方向反转。向量 −a 与 a 长度相同、方向相反。两个向量平行,当且仅当一个向量是另一个向量的标量倍数。

Scalar multiplication is the key tool for proving that lines are parallel in vector geometry problems.

数乘是向量几何题中证明直线平行的核心工具。

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