3.11.1 Vector 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
- $r = a + λ b + μ c$
  - 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?

Given the equation of a plane $r = (\binom{a_{1}}{\begin{matrix} a_{2} \\ a_{3} \end{matrix}}) + λ (\binom{b_{1}}{\begin{matrix} b_{2} \\ b_{3} \end{matrix}}) + μ (\binom{c_{1}}{\begin{matrix} c_{2} \\ c_{3} \end{matrix}})$ then the point r with position vector $(\binom{x}{\begin{matrix} y \\ z \end{matrix}})$ is on the plane if there exists a value of λ and μ such that

- $(\binom{x}{\begin{matrix} y \\ z \end{matrix}}) = (\binom{a_{1}}{\begin{matrix} a_{2} \\ a_{3} \end{matrix}}) + λ (\binom{b_{1}}{\begin{matrix} b_{2} \\ b_{3} \end{matrix}}) + μ (\binom{c_{1}}{\begin{matrix} c_{2} \\ c_{3} \end{matrix}})$
- This means that there exists a single value of λ and μ that satisfy the three parametric equations:
  - $x = a_{1} + λ b_{1} + μ c_{1}$
  - $y = a_{2} + λ b_{2} + μ c_{2}$
  - $z = a_{3} + λ b_{3} + μ c_{3}$
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 $a = 3 i + 2 j - k$ , $b = i - 2 j + 4 k$ , and $c = 4 i - j + 3 k$ 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
- $a x + b y + c z = d$
- 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 $(\begin{matrix} x \\ y \\ z \end{matrix})$ and a and n are both known vectors given in the form $(\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix})$ and $(\begin{matrix} n_{1} \\ n_{2} \\ n_{3} \end{matrix})$ then the Cartesian equation of the plane can be found using:
- $n \cdot r = n_{1} x + n_{2} y + n_{3} z$
- $a \cdot n = a_{1} n_{1} + a_{2} n_{2} + a_{3} n_{3}$
- Therefore $n_{1} x + n_{2} y + n_{3} z = a_{1} n_{1} + a_{2} n_{2} + a_{3} n_{3}$
- This simplifies to the form $a x + b y + c z = d$

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

To find $a x + b y + c z = d$ given $r = a + λ b + μ c$ :

Let $n = (\begin{matrix} a \\ b \\ c \end{matrix}) = b \times c$ then $d = n \cdot a$

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 $Π$ contains the point A $(2, 6, - 3)$ and has a normal vector $(\begin{matrix} 3 \\ - 1 \\ 4 \end{matrix})$ .

Find the equation of the plane in its Cartesian form.

JMMFvkWx_3-11-1-ib-aa-hl-vector-plane-cartesian-we-solution-a

Determine whether point B with coordinates

(- 1, 0, - 2)

lies on the same plane.

ghYsvxS~_3-11-1-ib-aa-hl-vector-plane-cartesian-we-solution-b

DP IB Maths: AA HL

Revision Notes

3.11.1 Vector Equations of Planes

Equation of a Plane in Vector Form

How do I find the vector equation of a plane?

How do I determine whether a point lies on a plane?

Exam Tip

Worked Example

Equation of a Plane in Cartesian Form

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

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

Exam Tip

Worked Example

Author: Amber

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