📚 Position Vectors | 位置向量
In Edexcel A-Level Mathematics, position vectors give a precise way to locate a point relative to a fixed origin. They connect coordinate geometry with vector algebra, and they appear regularly in pure mathematics questions on lines, triangles, parallelograms and 3D geometry.
在 Edexcel A-Level 数学中，位置向量提供了一种相对于固定原点精确定位点的方法。它将坐标几何与向量代数联系起来，在纯数学中关于直线、三角形、平行四边形和三维几何的问题中经常出现。
1. What is a Position Vector? | 什么是位置向量？
If a fixed origin O is chosen, the position vector of a point P is the vector OP. It is usually written as p or r, so p = →OP.
如果选定固定原点 O，点 P 的位置向量就是向量 OP。它通常写作 p 或 r，即 p = →OP。
In two dimensions, if P has coordinates (x, y), then its position vector is p = xi + yj, where i and j are the unit vectors along the x-axis and y-axis.
在二维空间中，如果 P 的坐标为 (x, y)，那么它的位置向量为 p = xi + yj，其中 i 和 j 分别是沿 x 轴和 y 轴的单位向量。
In three dimensions, the position vector extends to p = xi + yj + zk.
在三维空间中，位置向量扩展为 p = xi + yj + zk。
2. Position Vectors from Coordinates | 从坐标得到位置向量
Any coordinate point can be converted into a position vector by using the coordinates as the scalar components of i, j and k. For example, the point A(2, -3) has position vector a = 2i – 3j.
任何坐标点都可以通过将坐标作为 i、j 和 k 的标量分量来转换为位置向量。例如，点 A(2, -3) 的位置向量为 a = 2i – 3j。
When a vector is given in column form, it expresses the same information. The vector (2, -3) is identical to 2i – 3j, and both mean ‘move 2 units in the positive x-direction and 3 units in the negative y-direction’.
当向量以列形式给出时，它表达相同的信息。向量 (2, -3) 与 2i – 3j 相同，两者都表示 ‘沿 x 轴正方向移动 2 个单位、沿 y 轴负方向移动 3 个单位’。
| Point | Position vector |
|---|---|
| A(3, 4) | a = 3i + 4j |
| B(5, -2) | b = 5i – 2j |
| C(-1, 0) | c = -i |
3. Vector Between Two Points | 两点之间的向量
One of the most important results is that the vector from A to B is found by subtracting the position vector of A from the position vector of B.
最重要的结论之一是，从 A 到 B 的向量可以通过用 B 的位置向量减去 A 的位置向量得到。
→AB = b − a
For example, if a = 2i + j and b = 5i + 4j, then →AB = (5 − 2)i + (4 − 1)j = 3i + 3j.
例如，如果 a = 2i + j，b = 5i + 4j，那么 →AB = (5 − 2)i + (4 − 1)j = 3i + 3j。
This subtraction is essential: reversing the order gives the opposite direction, →BA = a − b.
这种减法很关键：颠倒顺序会得到相反方向，→BA = a − b。
4. Magnitude and Distance | 模长与距离
The magnitude of a position vector p = xi + yj is |p| = √(x² + y²). In 3D, |p| = √(x² + y² + z²).
位置向量 p = xi + yj 的模长为 |p| = √(x² + y²)。在三维中，|p| = √(x² + y² + z²)。
|p| = √(x² + y²)
The distance between two points A and B is simply the magnitude of →AB: distance AB = |b − a|.
两点 A 和 B 之间的距离就是 →AB 的模长：距离 AB = |b − a|。
Worked example: A(1, 2) and B(4, 6) give →AB = 3i + 4j, so distance AB = √(3² + 4²) = √25 = 5.
示例：A(1, 2) 和 B(4, 6) 得到 →AB = 3i + 4j，因此距离 AB = √(3² + 4²) = √25 = 5。
5. Unit Vectors | 单位向量
A unit vector has magnitude 1. To find the unit vector in the direction of any non-zero vector p, divide p by its magnitude.
单位向量的模长为 1。要求任何非零向量 p 方向上的单位向量，用 p 除以其模长。
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