IB DP Maths: AA HL复习笔记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
    • 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