Parametric form describes each coordinate of a line using a parameter, while Cartesian form eliminates that parameter and directly relates the coordinates. They represent exactly the same line but emphasize different information.
This is Mathematics: Analysis and Approaches HL content from AHL 3.14: Vector equation of a line and can be assessed on Paper 1 or Paper 2.
A line through the point , with direction vector , has vector equation
Equating corresponding components gives the parametric equations:
Rearranging each equation for and equating the results gives Cartesian form:
Conversion is therefore an elimination process. From parametric form, isolate in each component equation and equate the three expressions. To reverse the process, set the common Cartesian ratio equal to , then rearrange to obtain the three coordinate equations. The fixed terms identify a point on the line, and the coefficients of form a direction vector.
The choice of form depends on the task. Parametric form is especially useful for testing whether two lines intersect: equate their coordinates and solve for their parameters. Cartesian form is compact for reading coordinate restrictions and checking whether direction vectors are scalar multiples.
| Feature | Parametric form | Cartesian form |
|---|---|---|
| Standard form | , , | |
| Main advantage | Generates points by changing | Shows coordinate relationships without a parameter |
| Best used for | Intersections and motion | Recognizing or comparing lines |
For example, the line through with direction vector has parametric form
and Cartesian form
A common misconception is that Cartesian form contains no direction vector. It does: the denominators give its components. If a direction component is zero, do not divide by zero; for example, gives .
When asked to find either form, show the position vector and direction vector first. Check that your Cartesian denominators match the direction vector and remember that any non-zero scalar multiple represents the same direction.
