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

Equation of a Plane in Vector Form

How do I find the vector equation of a plane?

• The formula for finding the vector equation of a plane is

  • is the position vector of any point on the plane

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

  • and are two non-parallel direction vectors which are parallel to the plane

  •  and are scalars

• A plane in often denoted using the capital Greek letter

• There are an infinite number of ways to write the equation

  • There are an infinite number of options for

  • There are an infinite number of pairs non-parallel direction vectors

  • Any scalar multiple of a direction vector is also a direction vector

How do I find the vector equation of a plane that passes through three points?

• Suppose a line passes through the points with position vectors , and

• Find two direction vectors

  • For example, , or

    • It is important that all three points do not lie on the same line

• Use the given formula

  • For example,

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

• Write the components of the vectors in the equation

  • 

• Write the components of the position vector of the point to test

  •  

• Form a system of linear equations

  • 

  • 

  • 

• Solve two of the equations to find a value of 

• Check that these values also satisfy the third equation

  • If they do, then the point lies on the line

  • Otherwise, the point does not lie on the line