The Vector Equation of a Line

One point and one direction pin down a line. Write it as $\mathbf{r}=\mathbf{a}+\lambda\mathbf{d}$ , slide along it by changing $\lambda$ , test whether points lie on it, and find the angle between two lines from their directions alone.

Cambridge 9709 Paper 3Edexcel YMA01 WMA14AQA 7367 Paper 1≈ 35 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Write the vector equation of a line through a point with a given direction, or through two points.
Find points on a line, and decide whether a given point lies on it.
Recognise when two lines are parallel.
Find the acute angle between two lines.

Pause and think

A laser starts at $(1,2,3)$ and every second its beam front moves $(3,-2,2)$ . Where is it after $2$ seconds? After $t$ seconds?

Section 2 of 14

Core idea

Start hereStart here, go that way

A ship leaves port $(1,2,0)$ heading in direction $(2,-3,0)$ . Where is it after " $\lambda$ units of travel"? At $(1,2,0)+\lambda(2,-3,0)$ . Every point on its path is a starting point plus some multiple of a direction: that's the vector equation of a line.

The big idea

$\mathbf r=\mathbf a+\lambda\mathbf d$ : $\mathbf a$ a point on the line, $\mathbf d$ its direction, $\lambda$ any real number. The angle between lines uses only their directions (take the acute angle).

1. The vector equation

r = (1, 1) + λ(2, 1): each λ is one point

λ = -1.60

Positive

\lambda

moves forwards along

\mathbf d

, negative

\lambda

moves backwards. All of them together make the whole line.

\mathbf r=\mathbf a+\lambda\mathbf d

$\mathbf r$: position vector of a general point on the line.
$\mathbf a$: position vector of a known point.
$\mathbf d$: direction vector.
$\lambda$: parameter: each value gives one point.

Worked exampleLine through $A(1,2,-1)$ , $B(3,-1,4)$

Find its equation; is $(7,-7,14)$ on it?

Goal: Equation; yes/no.
Strategy: Direction $\overrightarrow{AB}$ ; solve for $\lambda$ in one component, check the others.

$\mathbf d=(2,-3,5)$
$\mathbf b-\mathbf a$ .
$\mathbf r=(1,2,-1)+\lambda(2,-3,5)$
Equation.
$1+2\lambda=7\Rightarrow\lambda=3$
$x$ -component.
$y=2-9=-7\ \checkmark,\quad z=-1+15=14\ \checkmark$
Check the others.

AnswerYes, at

\lambda=3

✓ CheckAll three components agree ✓.

Edge caseNot unique Any point on the line and any non-zero multiple of the direction give the same line.

2. Angle between lines

Worked exampleDirections $(1,-1,2)$ and $(1,1,-1)$

Find the acute angle between the lines.

Goal: Angle.
Strategy: Scalar product of directions; take modulus for acute.

$1-1-2=-2$
Dot product.
$\cos\theta=\frac{|-2|}{\sqrt6\sqrt3}=0.471$
Modulus gives the acute angle.

Answer

61.9^\circ

✓ CheckObtuse version

118.1^\circ

; acute requested ✓.

Pause and think

Line through $(0,0,5)$ parallel to the $x$ -axis?

Summary

$\mathbf r=\mathbf a+\lambda\mathbf d$ .
Angle from directions only, acute.

Direction vector: Gives the line's direction.

HistoryAhead of his time. In 1844 Hermann Grassmann, a German schoolteacher, published a book describing lines, planes and higher-dimensional spaces with what we now call vectors. Almost nobody understood it, and he never got the university post he hoped for. Decades later mathematicians realised he had laid the foundations of linear algebra.

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