Magnitude and Direction | 向量的大小与方向

📚 Magnitude and Direction | 向量的大小与方向

In A-Level Mathematics and Mechanics, a vector is a quantity that has both magnitude and direction, unlike a scalar which has only magnitude. This article explains how to calculate the magnitude and direction of a vector, how to convert between component form and magnitude-direction form, and how these ideas apply to forces and velocities.

在 A-Level 数学和力学中，向量是既有大小又有方向的量，这与只有大小的标量不同。本文将解释如何计算向量的大小和方向，如何在分量形式与大小-方向形式之间进行转换，以及这些概念如何应用于力和速度。

1. Vector notation and components | 向量表示与分量

A vector in two dimensions can be written as a = x i + y j, where i and j are the standard unit vectors along the positive x-axis and y-axis. The numbers x and y are called the components of the vector.

二维中的向量可以写作 a = x i + y j，其中 i 和 j 分别是沿正 x 轴和正 y 轴的标准单位向量。数 x 和 y 称为向量的分量。

For example, a = 3i + 4j means the vector has a horizontal component 3 and a vertical component 4. The same vector can be shown as a column vector with entries 3 and 4.

例如，a = 3i + 4j 表示该向量的水平分量为 3，竖直分量为 4。同一个向量也可表示为列向量，其元素为 3 和 4。

2. Magnitude of a vector | 向量的大小（模）

The magnitude of a vector a = x i + y j is written as |a| and is found using Pythagoras’ theorem. It represents the length of the directed line segment.

向量 a = x i + y j 的大小写作 |a|，利用勾股定理求得。它表示有向线段的长度。

|a| = √(x² + y²)

For example, the magnitude of a = 3i + 4j is |a| = √(3² + 4²) = √25 = 5.

例如，a = 3i + 4j 的大小为 |a| = √(3² + 4²) = √25 = 5。

3. Direction of a vector | 向量的方向

The direction of a vector is usually described by the angle θ that the vector makes with the positive x-axis, measured anticlockwise. The tangent of this angle is the ratio of the vertical component to the horizontal component.

向量的方向通常用它与正 x 轴所成的角 θ 来描述，按逆时针测量。该角的正切等于竖直分量与水平分量之比。

tan θ = y / x

To find θ, use θ = tan⁻¹(y / x). However, you must check the quadrant because tan⁻¹ only gives angles between -90° and 90°. Adjust by adding 180° if the vector points into the second or third quadrant.

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