📚 Position Vectors | 位置向量
Position vectors are the backbone of coordinate geometry and kinematics, linking algebra to spatial relationships. In A-Level Edexcel Mathematics, you will use position vectors to describe points, lines, and motion in both two and three dimensions. Mastering this concept is essential for solving vector equations, proving geometric properties, and tackling mechanics problems involving displacement, velocity and acceleration.
位置向量是坐标几何和运动学的支柱，它将代数与空间关系联系起来。在 A-Level Edexcel 数学中，你将使用位置向量描述点、直线以及二维和三维运动。掌握这一概念对于求解向量方程、证明几何性质以及处理涉及位移、速度和加速度的力学问题至关重要。
1. Definition of a Position Vector | 位置向量的定义
A position vector is a vector that starts from a fixed origin O and ends at a point P. It is denoted by the arrow OP or more commonly by a bold lowercase letter such as p. If P has coordinates (x, y) in 2D, then the position vector of P is p = xi + yj. In three dimensions the extension is natural: for P(x, y, z), p = xi + yj + zk.
位置向量是从固定原点 O 出发，终点为点 P 的向量。它用箭头 OP 或更常见的粗体小写字母如 p 表示。如果在二维中 P 坐标为 (x, y)，那么 P 的位置向量就是 p = xi + yj。三维中的推广很自然：对 P(x, y, z)，p = xi + yj + zk。
The fundamental idea is that the coordinates of a point are precisely the components of its position vector relative to the origin. This equivalence allows us to translate geometric problems into algebraic vector equations, making calculations systematic and powerful.
核心思想是，一个点的坐标正是它相对于原点的位置向量的分量。这种等价性使得我们可以把几何问题转化为代数的向量方程，让计算变得系统且强大。
2. Notation and Vector Components | 符号与向量分量
Position vectors are normally written in bold print, for example a, b, r. In handwritten work you would underline them or place an arrow above. The column vector form writes the components vertically: the point (3, -2) has position vector 3i - 2j, which can also be expressed as the column [3; -2]. Both notations are used in Edexcel examinations.
位置向量通常写成粗体，例如 a、b、r。手写时你可以在字母下画线或上方加箭头。列向量形式将分量垂直书写：点 (3, -2) 的位置向量为 3i - 2j，也可表示为列向量 [3; -2]。两种记法在爱德思考试中都会出现。
Fluency in switching between i, j form and column form is expected. You should also recognise that the zero vector 0 corresponds to the origin itself. Every vector in the plane or in space can be built as a combination of the standard basis vectors i, j and k.
要求在 i、j 形式与列形式之间熟练切换。你还需要知道零向量 0 对应着原点本身。平面或空间中的每一个向量都可以由标准基向量 i、j 和 k 组合而成。
3. Vector Between Two Points | 两点之间的向量
Given two points A and B with position vectors a and b, the vector directed from A to B is AB = b - a. This is one of the most frequently used identities in vector geometry. It follows directly from the triangle law: OA + AB = OB, so AB = OB - OA = b - a.
已知两点 A 和 B 的位置向量为 a 和 b，则由 A 指向 B 的向量为 AB = b - a。这是向量几何中最常用的恒等式之一，直接由三角形法则得出：OA + AB = OB，故 AB = OB - OA = b - a。
This relation is valid in any number of dimensions and is the basis for finding lines, checking collinearity, and calculating distances. The reverse vector from B to A is simply a - b, which is the negative of AB. Practise rewriting geometric statements such as 'ABCD is a parallelogram' in terms of position vectors: AB = DC leads to b - a = c - d.
这个关系在任何维度都成立，是求直线、检验共线性和计算距离的基础。由 B 指向 A 的反向向量就是 a - b，即 AB 的负向量。要练习用位置向量重写几何命题，例如“ABCD 是平行四边形”可得 b - a = c - d。
4. Magnitude of a Position Vector | 位置向量的模
The magnitude (or length) of a position vector r = xi + yj is |r| = √x² + y². It represents the straight-line distance from the origin to the point. For a 3D vector r = xi + yj + zk, the magnitude becomes √x² + y² + z². This is an extension of Pythagoras' theorem into three dimensions.
位置向量 r = xi + yj 的模（或长度）为 |r| = √x² + y²。它表示原点到该点的直线距离。对于三维向量 r = xi + yj + zk，模变为 √x² + y² + z²。这是勾股定理在三维中的推广。
In exam questions, you will often need the distance between two points A and B, which is simply |AB| = |b - a|. Always take the square root of the sum of the squared differences in each coordinate. A common mistake is to forget the final square root.
在考试题中，你经常需要求两点 A 和 B 的距离，它就是 |AB| = |b - a|。务必取各坐标差的平方和的平方根。一个常见错误是忘记最后的开平方根。
5. Midpoint and Section Formula | 中点与分割公式
If point P has position vector p and Q has position vector q, the midpoint M of PQ has position vector given by
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