Edexcel A Level Further Maths: Core Pure:复习笔记6.2.1 Equations of planes

Equation of a Plane in Vector Form

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

Exam Tip

  • The formula for the vector equation of a plane is given in the formula booklet, make sure you know what each part means
  • Be careful to use different letters, e.g. λ and μ as the scalar multiples of the two direction vectors

Worked Example

The points A, B and C have position vectors  respectively, relative to the origin O.

(a) Find the vector equation of the plane.

3-11-1-ib-aa-hl-vector-plane-vector-form-we-solution-a

(b) Determine whether the point D with coordinates (-2, -3, 5) lies on the plane.

3-11-1-ib-aa-hl-vector-plane-vector-form-we-solution-b

Equation of a Plane in Cartesian Form

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

Exam Tip

  • In an exam, using whichever form of the equation of the plane to write down a normal vector to the plane is always a good starting point

Worked Example

A plane  has equation . Find the equation of the plane in its Cartesian form.

al-fm-6-2-1-equation-of-plane-in-cartesian-form-we-solution

 

 

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