The Vector Equation of a Line — Mathematics

Introduction

The vector equation of a line is one of the most fundamental tools in the 3D geometry section of Pure Mathematics 3. It provides a compact, elegant way to describe every point on a straight line in two or three dimensions using vectors. In the 9709 exam, questions on this topic typically ask you to write down or derive the equation of a line given either a point and a direction, or two points — and then use that equation in further calculations involving intersections, angles, or distances. Mastering the form $\mathbf{r} = \mathbf{a} + t\mathbf{b}$ and understanding exactly what each symbol means is the essential first step.

Core Concept

The General Idea

A straight line in space is uniquely determined by:

a point it passes through, and
the direction in which it travels.

The vector equation encodes both pieces of information directly.

The Form $\mathbf{r} = \mathbf{a} + t\mathbf{b}$

Symbol	Name	Meaning
$\mathbf{r}$	Position vector	The position vector $\overrightarrow{OP}$ of a general point $P$ on the line
$\mathbf{a}$	Fixed position vector	The position vector of a known, fixed point $A$ on the line
$\mathbf{b}$	Direction vector	A vector parallel to the line; sets the direction of travel
$t$	Scalar parameter	A real number ( $t \in \mathbb{R}$ ); varying $t$ moves $P$ along the entire line

The equation says: "Start at point $A$ , then travel a scalar multiple $t$ of the direction vector $\mathbf{b}$ to reach any point $P$ on the line."

As $t$ ranges over all real numbers, $P$ traces out the complete infinite line:

$t = 0$ gives point $A$ itself.
$t > 0$ moves in the direction of $\mathbf{b}$ .
$t < 0$ moves in the direction opposite to $\mathbf{b}$ .

What Counts as a Direction Vector?

Any non-zero vector parallel to the line is a valid direction vector. In particular:

If the line passes through $A$ and $B$ , then $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$ is a direction vector.
Scalar multiples of a direction vector (e.g. $2\mathbf{b}$ or $-\mathbf{b}$ ) give the same line, just parametrised differently.

Important: The vector equation of a line is not unique. You may choose any point on the line as $\mathbf{a}$ and any parallel vector as $\mathbf{b}$ . Two different-looking equations can represent the same line.

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