Understand and use position vectors; calculate the distance between two points represented by position vectors

Position vectors are a fundamental concept in A-Level Mathematics, providing a way to describe the location of a point relative to a fixed origin O. Unlike displacement vectors, which describe movement from one point to another, a position vector is always anchored at the origin. For a point P, its position vector is denoted as \(\vec{OP}\) or simply \(\mathbf{p}\). In 2D, this is written as \(\mathbf{p} = \begin{pmatrix} x \\ y \end{pmatrix}\) or \(x\mathbf{i} + y\mathbf{j}\); in 3D, \(\mathbf{p} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\) or \(x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\). Understanding position vectors is essential for solving problems in geometry, kinematics, and vector algebra.

The distance between two points represented by position vectors is a direct application of the magnitude of a vector. If points A and B have position vectors \(\mathbf{a}\) and \(\mathbf{b}\), then the vector from A to B is \(\vec{AB} = \mathbf{b} - \mathbf{a}\). The distance between A and B is the magnitude of this vector: \(|\vec{AB}| = |\mathbf{b} - \mathbf{a}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) in 2D (with a similar extension to 3D). This formula is essentially the Pythagorean theorem in coordinate form and is a cornerstone for many geometric and physical calculations.

This topic is crucial for later work in mechanics (e.g., displacement, velocity, and acceleration vectors), further pure mathematics (e.g., vector equations of lines and planes), and even physics. Mastering position vectors and distance calculations builds a strong foundation for vector geometry and prepares students for more complex applications like finding the shortest distance between skew lines or solving problems involving forces and motion.