1. 55: The Vector Equation of a Plane
“Teach A Level Maths” Vol. 2: A2 Core Modules
55: The Vector Equation of a Plane
© Christine Crisp

2. There are 3 forms of the equation of a plane
There are 3 forms of the equation of a plane. We are going to look at 2 of them.
Suppose we have a vector n through a point A.
There is only one plane through A that is perpendicular to the vector. A x n

3. There are 3 forms of the equation of a plane
There are 3 forms of the equation of a plane. We are going to look at 2 of them.
Suppose we have a vector n through a point A. R x
Suppose R is any point on the plane ( other than A ). A x
Then, n
This is the equation of the plane since it is satisfied by the position vector of any point on the plane, including A.
The scalar product can be expanded to give

4. e. g. 1 Find the equation of the plane through the point A(2, 3, -1)
e.g.1 Find the equation of the plane through the point A(2, 3, -1) perpendicular to
Solution:

5. Calculating the left-hand scalar product gives the Cartesian form of the equation.

6. Diagram

7. It is useful in some problems to know that a vector n will be perpendicular to the plane if it is perpendicular to 2 non-parallel vectors in the plane. A x C x
There are an infinite number of vectors perpendicular to AC. For example, one lies on the plane.

8. Others lie at angles to the plane.
It is useful in some problems to know that a vector n will be perpendicular to the plane if it is perpendicular to 2 non-parallel vectors in the plane. A x C x
There are an infinite number of vectors perpendicular to AC. For example, one lies on the plane.
Others lie at angles to the plane.
Only one is also perpendicular to AB.

9. Others lie at angles to the plane.
It is useful in some problems to know that a vector n will be perpendicular to the plane if it is perpendicular to 2 non-parallel vectors in the plane. A x C x B x
There are an infinite number of vectors perpendicular to AC. For example, one lies on the plane.
Others lie at angles to the plane.
Only one is also perpendicular to AB.
This one is perpendicular to the plane.

10. e.g.2 Show that the vector n is perpendicular to the plane containing the points A, B and C where
The plane containing A, B and C also contains the vectors and
Solution:

11. So,
n is perpendicular to 2 vectors in the plane so is perpendicular to the plane.

12. n is a vector perpendicular to the plane and
SUMMARY
The vector equation of a plane is given by or
where a is the position vector of a fixed point on the plane
n is a vector perpendicular to the plane and
r is the position vector of any point on the plane.
The Cartesian form is
where n1, n2 and n3 are the components of n and
n is called the normal vector

13. Exercise
1. Find a vector equation of the plane through the point with normal vector
2. Find the Cartesian equation of the plane through the point perpendicular to the vector

14. 1.
Plane through the point with normal vector
Solution:

15. Diagram

16. 2. Find the Cartesian equation of the plane through the point perpendicular to the vector
Solution:

17. Exercise
3. Show that is perpendicular to the plane
containing the points A(1, 0, 2), B(2, 3, -1) and C(2, 2, -1 ).
Solution:
n is perpendicular to 2 vectors in the plane so is perpendicular to the plane.

18. Perpendicular Distance of a Plane from the Origin
Using the unit normal the plane equation becomes
= |a|1cos
= d d
where d is the perpendicular distance of the plane from the origin

19. Perpendicular Distance of a Plane from the Origin
Using the unit normal the plane equation becomes
= |a|1cos
= d d
So if the plane equation is converted into the unit normal plane equation by dividing by the magnitude on n then d represents the perpendicular distance of a plane from the origin

20. Find the distance of the plane below from the origin
So the unit normal form is
Perpendicular distance of the plane from the origin =

21. Perpendicular Distance Between Two Planes
Find the perpendicular distance of each plane from the origin and hence find the distance between the two planes by subtraction.
Find the perpendicular distance between
The negative sign means that plane 2 is on the other side of the origin from plane 1.

22. Plane 1
Plane 2
The perpendicular distance between the planes =
Diagram