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
    • 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?

• 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
  • 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