Representing Vectors | 向量的表示方法

📚 Representing Vectors | 向量的表示方法

Vectors describe quantities that have both magnitude and direction. In A-Level Mathematics, you must be able to switch confidently between column vectors, unit vector form, position vectors and geometric diagrams.

向量描述的是同时具有大小和方向的量。在 A-Level 数学中，你必须能够在列向量、单位向量形式、位置向量和几何图形之间熟练转换。

1. What Is a Vector? | 什么是向量？

A vector is a quantity with both magnitude and direction, such as displacement, velocity or force. A scalar has only magnitude, such as speed, distance or mass.

向量是同时具有大小和方向的量，例如位移、速度或力。标量只有大小，例如速率、距离或质量。

A vector can be shown geometrically as a directed line segment. The length of the arrow represents the magnitude, and the arrowhead shows the direction.

向量在几何上可以表示为一条有向线段。箭头的长度表示大小，箭头指向表示方向。

2. Column Vector Notation | 列向量表示法

In two dimensions, a vector with horizontal component x and vertical component y is written as a column vector. The top entry is the i-component and the bottom entry is the j-component.

在二维空间中，水平分量为 x、垂直分量为 y 的向量写成列向量。上方的分量是 i 分量，下方的分量是 j 分量。

a = (xy)

For example, a displacement of 3 units right and 4 units up is written as a = (3 4) with 3 above 4 in a column.

例如，向右 3 个单位、向上 4 个单位的位移写作 a = (3 4)，其中 3 在 4 的上方组成一列。

Column vectors make addition and scalar multiplication easy because each component is kept separate.

列向量使加法和标量乘法变得简单，因为每个分量都分别处理。

3. Unit Vectors i and j | 单位向量 i 与 j

The standard unit vectors in two dimensions are i and j, where i has length 1 in the x-direction and j has length 1 in the y-direction.

二维空间中的标准单位向量是 i 和 j，其中 i 在 x 方向长度为 1，j 在 y 方向长度为 1。

i = (10), j = (01)

Any two-dimensional vector can be expressed as a = xi + yj. This is often called component form or unit vector form.

任何二维向量都可以表示为 a = xi + yj。这通常称为分量形式或单位向量形式。

In Edexcel questions, you are expected to move between column vector form and i, j form without changing the vector.

在 Edexcel 的考题中，你需要能够在列向量形式和 i、j 形式之间转换，而不改变向量本身。

4. Position Vectors | 位置向量

A position vector starts at the origin O and ends at a point P. If P has coordinates (x, y), then the position vector of P is OP = xi + yj.

位置向量从原点 O 出发，终点为某一点 P。如果 P 的坐标为 (x, y)，那么 P 的位置向量为 OP = xi + yj。

OP = (xy)

Position vectors are useful because any vector between two points can be found by subtracting their position vectors: AB = b – a.

位置向量非常有用，因为两点之间的任何向量都可以通过它们的位置向量相减得到：AB = b – a。

Here a and b are the position vectors of A and B respectively. This rule is central in coordinate geometry and vector geometry.

这里 a 和 b 分别是 A 和 B 的位置向量。这一规则在坐标几何和向量几何中非常核心。

5. Magnitude

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