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 direction vectors must be
- The formula for finding the vector equation of a plane is
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- 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
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- 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
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- 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
Examiner 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. and as the scalar multiples of the two direction vectors
Worked example
The points A, B and C have position vectors , , and respectively, relative to the origin O.
(a) Find the vector equation of the plane.
(b) Determine whether the point D with coordinates (-2, -3, 5) lies on the plane.