Vector Equation of a Line, Intersections

IB Mathematics AI HL· 6 min read

1. Vector Equation of a Line★★☆☆☆⏱ 15 min

📘 Definition

Vector Equation of a Line

$r = a + λ d, λ \in R$

For a line passing through a fixed point with position vector $a$ and parallel to direction vector $d$ , any point $r$ on the line is given by the sum of $a$ and a scalar multiple $λ$ of $d$ . The parameter $λ$ can take any real value, generating all points on the line.

Example:

A line through $(1, 2, 3)$ parallel to $2 - 1 0$ has equation $r = 123 + λ 2 - 1 0$

This form works for both 2D and 3D space — only the dimension of the position and direction vectors changes. Any scalar multiple of the direction vector produces an equivalent equation for the same line.

📐 Worked Example

Find the vector equation of the line passing through points $P (2, - 1, 4)$ and $Q (5, 2, 1)$ .

1
Choose the position vector of a fixed point on the line, using $P$ :
2
$\mathbf{a} = \begin{pmatrix}2 \\ -1 \\ 4\end{pmatrix}$
3
Calculate the direction vector as the vector between the two points:
4
$\mathbf{d} = \overrightarrow{PQ} = Q - P = \begin{pmatrix}5-2 \\ 2 - (-1) \\ 1 - 4\end{pmatrix} = \begin{pmatrix}3 \\ 3 \\ -3\end{pmatrix}$
5
Simplify the direction vector by dividing by the scalar 3 (any non-zero scalar multiple is valid):
6
$\mathbf{d} = \begin{pmatrix}1 \\ 1 \\ -1\end{pmatrix}$
7
Substitute into the general vector equation:
8
$\mathbf{r} = \begin{pmatrix}2 \\ -1 \\ 4\end{pmatrix} + \lambda \begin{pmatrix}1 \\ 1 \\ -1\end{pmatrix}, \lambda \in \mathbb{R}$

2. Converting Between Common Line Forms★★☆☆☆⏱ 15 min

For calculations, you will often need to convert the vector equation into parametric or Cartesian form. Starting from the general 3D vector equation:

\mathbf{r} = \begin{pmatrix}a_1 \\ a_2 \\ a_3\end{pmatrix} + \lambda \begin{pmatrix}d_1 \\ d_2 \\ d_3\end{pmatrix}

Parametric form: Split into separate equations for each coordinate: $x = a_{1} + λ d_{1}$ , $y = a_{2} + λ d_{2}$ , $z = a_{3} + λ d_{3}$
Cartesian form: Rearrange to isolate $λ$ from each equation, then equate: $\frac{x - a _{1}}{d _{1}} = \frac{y - a _{2}}{d _{2}} = \frac{z - a _{3}}{d _{3}}$

📐 Worked Example

Convert $r = 13 - 2 + λ 2 - 1 4$ to parametric and Cartesian form.

1
Let $r = x y z$ , equate components to get parametric form:
2
$x = 1 + 2\lambda, \quad y = 3 - \lambda, \quad z = -2 + 4\lambda$
3
Rearrange each equation to isolate $λ$ :
4
$\lambda = \frac{x-1}{2}, \quad \lambda = \frac{y-3}{-1}, \quad \lambda = \frac{z+2}{4}$
5
Equate all expressions to get Cartesian form:
6
$\frac{x-1}{2} = 3 - y = \frac{z+2}{4}$

3. Intersections and Relationships Between Two Lines★★★☆☆⏱ 20 min

📘 Definition

Relationships Between Two Lines in 3D

There are three possible cases for two lines in 3D space:

Parallel: Direction vectors are scalar multiples of each other, lines never meet.
Intersecting: Lines are not parallel and share exactly one common point.
Skew: Lines are not parallel and never meet, only possible in 3D.

To find if two lines intersect, write both in parametric form with different parameters, set coordinates equal, solve the system of equations, then check consistency.

📐 Worked Example

Determine if lines $L_{1} : r_{1} = 223 + λ 1 - 1 2$ and $L_{2} : r_{2} = 104 + μ 211$ intersect, and find the intersection point if they do.

1
Write parametric equations for both lines:
2
$L_1: x=2+\lambda,\ y=2-\lambda,\ z=3+2\lambda \\ L_2: x=1+2\mu,\ y=\mu,\ z=4+\mu$
3
Set coordinates equal to get a system of equations:
4
$2 + \lambda = 1 + 2\mu \implies \lambda - 2\mu = -1 \quad (1) \\ 2 - \lambda = \mu \implies \lambda + \mu = 2 \quad (2) \\ 3 + 2\lambda = 4 + \mu \implies 2\lambda - \mu = 1 \quad (3)$
5
Solve (1) and (2): Substitute $λ = 2 - μ$ from (2) into (1):
6
$(2 - \mu) - 2\mu = -1 \implies 2 - 3\mu = -1 \implies \mu = 1, \lambda = 1$
7
Check consistency with equation (3): Left side $= 2 (1) - 1 = 1$ , which matches the right side. The solution is consistent.
8
Substitute $λ = 1$ into $L_{1}$ to get the intersection point:
9
$x=3,\ y=1,\ z=5 \implies \text{Intersection at } (3, 1, 5)$

4. Common Pitfalls

Wrong move:

Using the same parameter variable for both lines when solving for intersections

Why:

Each line has an independent free parameter, so using the same variable incorrectly forces the parameter value to be the same on both lines

Correct move:

Use $λ$ for the first line and $μ$ for the second line to keep the parameters independent

Wrong move:

Stopping after solving two equations, not checking consistency with the third equation in 3D

Why:

Two equations will always produce a solution even for skew lines, so you must confirm the solution satisfies all equations

Correct move:

Always substitute the calculated parameter values into the third equation to check consistency before concluding lines intersect

Wrong move:

Claiming non-parallel lines that do not intersect are parallel

Why:

This ignores the third category of lines unique to 3D space

Correct move:

Classify non-parallel lines that never intersect as skew lines

Wrong move:

Forgetting that direction vectors can be any scalar multiple

Why:

Sign errors or incorrect scaling leads to incorrect conversion to Cartesian form

Correct move:

Simplify direction vectors to have integer components with a positive leading term for consistency

5. Quick Reference Cheatsheet

Concept	Rule/Formula
Vector equation of line	$r = a + λ d$
Parametric form (3D)	$x = a_{1} + λ d_{1}, y = a_{2} + λ d_{2}, z = a_{3} + λ d_{3}$
Cartesian form (3D)	$\frac{x - a _{1}}{d _{1}} = \frac{y - a _{2}}{d _{2}} = \frac{z - a _{3}}{d _{3}}$
Parallel lines	$d_{1} = k d_{2}$ for some scalar $k$
Intersecting lines	Non-parallel, consistent parameter solution exists
Skew lines	Non-parallel, no consistent parameter solution

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Find intersection of two 3D lines
2023 · 2
Classify lines as parallel/intersecting/skew

What's Next

Understanding the vector equation of a line is the core foundation for all further 3D vector topics in IB AI HL, including the vector equation of planes and finding intersections between lines and planes, which are frequently tested in both Paper 1 and Paper 2. This topic builds on your knowledge of coordinate geometry and vectors, and connects to applied problems involving paths of moving objects. Mastering the method for checking intersections and classifying lines will make all more complex 3D geometry problems much easier to approach and solve correctly.