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
    • therefore they will?intersect?at some point on the plane
• 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
    • s and t are scalars
• The formula can also be written as
  • 
    • Where?r?is the?position vector?of any point on the plane
    • a, b, c?are?the?position vectors?of known points on the plane
    • λ and μ are scalars
  • These formulae are?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
    • A version of this is?given?in the?formula booklet

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

• Given the equation of the plane
  • Form three equations
    • 
    • 
    • 
• Choose a pair of equations and use them to form an equation without μ
• Choose another pair and form another equation without μ
• Use your two expressions to form an equation without μ and λ
• Rewrite the equation in the form