IBMathsDPAnalysis & Approaches (AA)HLRevision Notes3. Geometry & TrigonometryVector Equations of LinesEquation of a Line in Parametric Form

Equation of a Line in Parametric Form (DP IB Analysis & Approaches (AA): HL): Revision Note

Written by: Amber

Reviewed by: Dan Finlay

Updated on 26 November 2025

Equation of a line in parametric form

What is the equation of a line in parametric form?

The parametric form of the equation of a line is:
- $x = x_{0} + λ l$
- $y = y_{0} + λ m$
- $z = z_{0} + λ n$
$(x_{0}, y_{0}, z_{0})$ is the coordinates of any point on the line
The vector $l i + m j + n k$ is a direction vector of the line

Examiner Tips and Tricks

This is given in the formula booklet under the geometry and trigonometry section. However, you need to remember what the components represent.

You can write it easily in its vector form $r = a + λ b$
- $(\binom{x}{\begin{matrix} y \\ z \end{matrix}}) = (\binom{x_{0}}{\begin{matrix} y_{0} \\ z_{0} \end{matrix}}) + λ (\binom{l}{\begin{matrix} m \\ n \end{matrix}})$
  - $r = (\binom{x}{\begin{matrix} y \\ z \end{matrix}})$
  - $a = (\binom{x_{0}}{\begin{matrix} y_{0} \\ z_{0} \end{matrix}})$
  - $b = (\binom{l}{\begin{matrix} m \\ n \end{matrix}})$

Worked Example

Write the parametric form of the equation of the line which passes through the point (-2, 1, 0) with direction vector $(\binom{3}{\begin{matrix} 1 \\ - 4 \end{matrix}})$ .

Answer:

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