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

Equation of a plane in cartesian form

How do I find the vector equation of a plane using a normal vector?

• The formula for finding the vector equation of a plane is $\mathit{r}·\mathit{n}=\mathit{a}\mathbf{·}\mathit{n}$

  • $\mathit{r}$ is the position vector of any point on the plane

  • $\mathit{a}$ is the position vector of a known point on the plane

  • $\mathit{n}$ is a normal vector which is perpendicular to the plane

• A normal vector can be found by taking the vector product of two non-parallel vectors which are parallel to the plane

  • For $\mathit{r}=\mathit{a}+\lambda \mathit{b}+\mu \mathit{c}$ use $\mathit{n}=\mathit{b}\times \mathit{c}$

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

• The formula for finding the Cartesian equation of a plane is $ax+by+cz=d$

  • The vector $\left(\begin{array}{c}a\\ b\\ c\end{array}\right)$ is perpendicular to the plane

• Cartesian equations represent the same planes if they are scalar multiples of each other

  • For example, $2x+3y+z=5$ and $4x+6y+2z=10$ represent the same plane

  • For example, $2x+3y+z=5$ and $4x+6y+2z=9$ represent different planes

How do I find the equation of a plane in Cartesian form given the vector form?

• Suppose you are given the equation $\mathit{r}=\mathit{a}+\lambda \mathit{b}+\mu \mathit{c}$

• STEP 1
  Use the vector product to find a normal vector to the plane

  • $\mathit{n}=\mathit{b}\times \mathit{c}$

• STEP 2
  Write the expression $ax+by+cz$

  • $\left(\begin{array}{c}a\\ b\\ c\end{array}\right)=\mathit{n}$

• STEP 3
  Use the scalar product of the normal vector and the position vector to find the value of $d$

  • $d=\mathit{n}·\mathit{a}$

How do I determine whether a point lies on a plane?

• You can do this easily using the Cartesian equation

• Substitute the coordinates of the point into the Cartesian equation

  • If the equation is satisfied, then the point lies on the plane

  • Otherwise, it does not lie on the plane