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

Equation of a Line in Cartesian Form

What is the equation of a line in Cartesian form?

• The Cartesian form of the equation of a line is

  • is the coordinates of any point on the line

  • The vector is a direction vector of the line

How do I find the equation of a line in Cartesian form?

• You can find the Cartesian form by converting from the parametric form

  • Rearrange each equation to make the subject

    • e.g.

  • Set the equations equal to each other

How do I convert from Cartesian form to vector form?

• STEP 1
  Set each part of the equation equal to individually

• STEP 2
  Rearrange each of these three equations to make , , and the subjects

  • This will give you the three parametric equations

    • 

    • 

    • 

• STEP 3
  Write this in the vector form 

• STEP 4
  Set   to equal 

What do I do if any of the components of the direction vector are zero?

• If one component of the direction vector is equal to zero

  • Then the corresponding variable is equal to a constant as it does not change

    • e.g.

  • You write the Cartesian equation using two separate equations

    • e.g.

• If two components of the direction vector are equal to zero

  • Then both corresponding variables are equal to constants

    • e.g. and

  • You write the Cartesian equation using three separate equations

  • You need to include the parameter for the third equation

    • e.g.