Equation of a Line in Cartesian Form (DP IB Analysis & Approaches (AA): HL): 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 $\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}$

  • $\left(x_0,y_0,z_0\right)$ is the coordinates of any point on the line

  • The vector $l\mathit{i}+m\mathit{j}+n\mathit{k}$ 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 $\lambda$ the subject

    • e.g. $x=x_0+\lambda l \Rightarrow \lambda =\frac{x-x_0}{l}$

  • 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 $\lambda$individually

• STEP 2
  Rearrange each of these three equations to make $x$, $y$, and $z$ the subjects

  • This will give you the three parametric equations

    • $x=x_0+\lambda l$

    • $y=y_0+\lambda m$

    • $z=z_0+\lambda n$

• STEP 3
  Write this in the vector form $\left(\genfrac{}{}{0pt}{}{x}{\begin{array}{c}y\\ z\end{array}}\right)= \left(\genfrac{}{}{0pt}{}{x_0}{\begin{array}{c}{y}_0\\ z_0\end{array}}\right)+\lambda \left(\genfrac{}{}{0pt}{}{l}{\begin{array}{c}m\\ n\end{array}}\right)$

• STEP 4
  Set $\mathit{r}$  to equal $\left(\genfrac{}{}{0pt}{}{x}{\begin{array}{c}y\\ z\end{array}}\right)$

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. $x=x_0$

  • You write the Cartesian equation using two separate equations

    • e.g. $x=x_0,\frac{y-y_0}{m}=\frac{z-z_0}{n}$

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

  • Then both corresponding variables are equal to constants

    • e.g. $x=x_0$ and $z=z_0$

  • You write the Cartesian equation using three separate equations

  • You need to include the parameter for the third equation

    • e.g. $x=x_0,z=z_0,\frac{y-y_0}{m}=\lambda$