The vector equation of a line: r = a + tb

By the end of this lesson

By the end you can say what $\mathbf{r}$ , $\mathbf{a}$ , $\mathbf{b}$ and $t$ each mean in $\mathbf{r} = \mathbf{a} + t\mathbf{b}$ , write the equation of a line from a point and a direction or from two points, and check whether a point lies on a line.

Describing a line with vectors

In 2D you describe a line with $y = mx + c$ . In 3D that doesn't work: there is no single "gradient", because a line can tilt in two ways at once. Vectors solve this neatly.

To pin down a line you need two things:

A point it passes through. That tells you where the line is.
A direction. That tells you which way it goes.

Think of giving directions to a friend: "Start at the post office, then walk north-east. Every place you can reach that way, forwards or backwards, is on the line."

What each symbol means

The vector equation of a line is

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

Symbol	Meaning
$\mathbf{a}$	the position vector of one fixed point $A$ on the line: the arrow from the origin $O$ to $A$
$\mathbf{b}$	a direction vector: any non-zero vector pointing along the line
$t$	a parameter: an ordinary number that you are free to choose, positive, negative or zero
$\mathbf{r}$	the position vector of the point you reach for that choice of $t$

Bold letters are vectors (in handwriting, underline them). A vector in 3D has three components, written as a column, such as $\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ , or as $2\mathbf{i} - \mathbf{j} + 3\mathbf{k}$ .

How to read it: start at $A$ (that's $\mathbf{a}$ ), then travel $t$ lots of $\mathbf{b}$ .

$t = 0$ gives $\mathbf{r} = \mathbf{a}$ : you stay at $A$ .
$t = 1$ gives $\mathbf{a} + \mathbf{b}$ : one step along the direction.
$t = 2$ is two steps; $t = -1$ is one step backwards; $t = 0.5$ is half a step.

As $t$ runs through every real number, $\mathbf{r}$ visits every point on the line, and no others.

Worked example: From a point and a direction

Find a vector equation of the line through the point $A(2, -1, 3)$ that is parallel to the vector $4\mathbf{i} - 2\mathbf{j} + \mathbf{k}$ .

Have a go first?

Check point and direction

One-line answerQuestion 1 · Recall

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