Working with Vectors | 向量运算与几何应用

📚 Working with Vectors | 向量运算与几何应用

Vectors are essential in Edexcel A-Level Mathematics because they describe quantities with both magnitude and direction, such as displacement, velocity and force. This revision guide focuses on the core skills for working with vectors: notation, magnitude, direction, unit vectors, addition, scalar multiplication, position vectors, parallel vectors, vector equations and geometric applications.

向量在 Edexcel A-Level 数学中至关重要，因为它们描述既有大小又有方向的量，例如位移、速度和力。本复习指南聚焦向量运算的核心技能：符号表示、大小、方向、单位向量、加法、数乘、位置向量、平行向量、向量方程以及几何应用。

1. Vector Notation and Representation | 向量符号与表示

A vector is often written in bold type, such as a, or with an arrow above the letter. In handwriting, you may underline the letter: a. A column vector in two dimensions can be written with components, for example a = (3, 4) or a = 3i + 4j.

向量通常用粗体字母表示，如 a，或在字母上方加箭头。手写时可在字母下方加下划线：a。二维列向量可用分量表示，例如 a = (3, 4) 或 a = 3i + 4j。

The component form breaks a vector into horizontal and vertical parts. For a = xi + yj, x is the i-component and y is the j-component. In three dimensions we add a k-component: a = xi + yj + zk.

分量形式将向量分解为水平分量和垂直分量。对于 a = xi + yj，x 是 i 分量，y 是 j 分量。在三维空间中，我们增加 k 分量：a = xi + yj + zk。

You should be comfortable converting between column vectors and i, j notation. For example, the vector (2, -5) is the same as 2i – 5j. Negative components simply point in the negative coordinate directions.

你应该能够熟练地在列向量和 i、j 表示法之间转换。例如向量 (2, -5) 等价于 2i – 5j。负分量只是指向负坐标方向。

a = xi + yj = (x, y)

2. Magnitude and Direction | 向量的大小与方向

The magnitude of a vector is its length. For a two-dimensional vector a = xi + yj, the magnitude is given by |a| = √(x² + y²). This comes from Pythagoras’ theorem.

向量的大小就是它的长度。对于二维向量 a = xi + yjPublished by TutorHao | A-Level Revision Series | aleveler.com

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