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

Equation of a Plane in Cartesian Form

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

• The cartesian equation of a plane is given in the form

  • 

  • This is given in the formula booklet

• A normal vector to the plane can be used along with a known point on the plane to find the cartesian equation of the plane

  • The normal vector will be a vector that is perpendicular to the plane

• The scalar product of the normal vector and any direction vector on the plane will the zero

  • The two vectors will be perpendicular to each other

  • The direction vector from a fixed-point A to any point on the plane, R can be written as r – a

  • Then n ∙ (r – a) = 0 and it follows that (n ∙ r) – (n ∙ a) = 0

• This gives the equation of a plane using the normal vector:

  • n ∙ r = a ∙ n

    • Where r is the position vector of any point on the plane

    • a is the position vector of a known point on the plane

    • n is a vector that is normal to the plane

  • This is given in the formula booklet

• If the vector r is given in the form  and a and n are both known vectors given in the form and  then the Cartesian equation of the plane can be found using:

  • 

  • 

  • Therefore 

  • This simplifies to the form 

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

• The Cartesian equation of a plane can be found if you know

  • the normal vector and

  • a point on the plane

• The vector equation of a plane can be used to find the normal vector by finding the vector product of the two direction vectors

  • A vector product is always perpendicular to the two vectors from which it was calculated

• The vector a given in the vector equation of a plane is a known point on the plane

  • Once you have found the normal vector then the point a can be used in the formula n ∙ r = a ∙ n to find the equation in Cartesian form

• To find given  :

  • Let  then