Vector Notation and Operations – Pure Mathematics 3 (9709) — Mathematics

Introduction

Vectors are quantities that have both magnitude and direction. In Pure Mathematics 3, vectors are used to describe positions and displacements in two and three dimensions, and they underpin later work on lines, planes, and intersections. The exam tests your ability to write vectors correctly in multiple notations, perform algebraic operations fluently, and interpret those operations geometrically — for example, recognising when four points form a parallelogram.

Core Concept

What is a vector?

A vector represents a displacement. The vector $\overrightarrow{AB}$ means "start at $A$ , travel to $B$ ." Its magnitude (length) is written $|\overrightarrow{AB}|$ .

A position vector locates a point relative to a fixed origin $O$ . The position vector of point $A$ is $\overrightarrow{OA}$ , often written as $\mathbf{a}$ .

Standard notations you must know

Notation	Meaning	Example
Column vector $\begin{pmatrix} x \\ y \end{pmatrix}$	2D vector with components $x$ , $y$	$\begin{pmatrix} 3 \\ -2 \end{pmatrix}$
Column vector $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$	3D vector with components $x$ , $y$ , $z$	$\begin{pmatrix} 1 \\ 0 \\ -4 \end{pmatrix}$
$x\mathbf{i} + y\mathbf{j}$	2D unit-vector form	$3\mathbf{i} - 2\mathbf{j}$
$x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$	3D unit-vector form	$\mathbf{i} - 4\mathbf{k}$
$\overrightarrow{AB}$	Displacement from $A$ to $B$	—
$\mathbf{a}$ (bold) or $\underline{a}$ (underline in handwriting)	Named vector	—

The unit vectors $\mathbf{i}$ , $\mathbf{j}$ , $\mathbf{k}$ point along the positive $x$ -, $y$ -, $z$ -axes respectively.

Finding $\overrightarrow{AB}$ from position vectors

\overrightarrow{AB} = \mathbf{b} - \mathbf{a} = \overrightarrow{OB} - \overrightarrow{OA}

This is one of the most frequently used results in the topic.

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