3.11.1 Vector Equations of Planes

How do I find the vector equation of a plane?

• A plane is a flat surface which is two-dimensional
  • Imagine a flat piece of paper that continues on forever in both directions
• A plane in often denoted using the capital Greek letter Π
• The vector form of the equation of a plane can be found using two direction vectors on the plane
  • The direction vectors must be
    • parallel to the plane
    • not parallel to each other
  • If both direction vectors lie on the plane then they will intersect at a point
• The formula for finding the vector equation of a plane is
  • 
    • Where r is the position vector of any point on the plane
    • a is the position vector of a known point on the plane
    • b and c are two non-parallel direction (displacement) vectors parallel to the plane
    • λ and μ are scalars
  • The formula is given in the formula booklet but you must make sure you know what each part means
• As a could be the position vector of any point on the plane and b and c could be any non-parallel direction vectors on the plane there are infinite vector equations for a single plane

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

• Given the equation of a plane  then the point r with position vector is on the plane if there exists a value of λ and μ such that

• • 
  • This means that there exists a single value of λ and μ that satisfy the three parametric equations:
    • 
    • 
    • 
• Solve two of the equations first to find the values of λ and μ that satisfy the first two equation and then check that this value also satisfies the third equation
• If the values of λ and μ do not satisfy all three equations, then the point r does not lie on the plane

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