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Equations of Planes

  2. Geometry & Trigonometry
  3. Equations of Planes
    4. 30'

This is a hugely important topic in the HL course. We can describe planes in 3D in a number of different ways. It is not enough to learn the different equations, but it is vital that you have a strong conceptual understanding of the different forms of the equations of the plane, in particular, the normal form. You will need to be familiar with the Vector Product and the Scalar Product before you start this topic. Questions in the exam on this topic are often long questions from section B.

Key Concepts

On this page, you should learn about

  • vector equations of planes in three dimensions in the three different forms
    • vector form
    • parametric form
    • cartesian form
 

Essentials

The following videos will help you understand all the concepts from this page

Planes - Vector, Normal & Cartesian Forms

In the following video we are going to look at the equations of planes, in particular the three different forms: Vector, Normal and Cartesian.

r = a + λb + μc Vector Form
r∙n = a∙n Normal Form
ax + by + cz = d Cartesian Form

We will find out how to convert from one form to another by looking at the follwing example:

Convert the following into normal and Cartesian form

\(\textbf{r} =\left( \begin{matrix} 1 \\ 2 \\ 0 \end{matrix} \right) +\lambda \left( \begin{matrix} 2 \\ 1 \\ 1 \end{matrix} \right) +\mu \left( \begin{matrix} 3 \\ 0 \\ -1 \end{matrix} \right) \)

Notes from the video

 

Equation of a Plane from 3 Points

In the following video we are going to try to find the equation of a plane that is formed by three points. Also, we will be able to see if a 4th point lies in the same plane (Are the four points coplanar?)

Find the equation of the plane formed by the triangle A(1 , 2 , -1) , B(2 , -2 , 3) and C(0 , 2 , 1)

Notes from video

Does a 4th point D (1 , -1 , 2) lie in the plane?

 
 

Summary

Print from here

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Test Yourself

Here is a short quiz to practise the skills you have learned in this topic.


START QUIZ!

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Exam-style Questions

Question 1

A plane has vector equation \(\textbf{ r }=\left( \begin{matrix} 1 \\ 2 \\ 0 \end{matrix} \right) +\mu \left( \begin{matrix} 2 \\ 1 \\ 1 \end{matrix} \right) +\lambda \left( \begin{matrix} 3 \\ 0 \\ -1 \end{matrix} \right) \)

Show that the Cartesian equation of the plane is x - 5y + 3z + 9 = 0

Hint

Full Solution

 

 

Question 2

The coordinates of points A, B and C are given as (5,4,1) , (5,1,-2) and (1,-1,2) respectively.

a) Find the equation of the plane that passes through A, B and C

b) Find the equation of the plane that is perpendicular to AB and passes through C

Hint

Full Solution

 

 

Question 3

A plane has vector equation \(\textbf{ r }=\left( \begin{matrix} 1 \\ 2 \\ 0 \end{matrix} \right) +\mu \left( \begin{matrix} -2 \\ 0 \\ 5 \end{matrix} \right) +\lambda \left( \begin{matrix} 0 \\ -4 \\ 5 \end{matrix} \right) \)

a) Find the Cartesian equation of the plane

b) The plane meets the x, y and z axes at A, B and C respectively. OABC forms a pyramid. Find the volume of the pyramid.

Hint

Full Solution

 

 

Question 4

Find the Cartesian equation of the plane that is perpendicular to the plane 2x - y + z = 8 and contains the points A(4,2,-3) and B(6,1,-1).

Hint

Full Solution

 

 
MY PROGRESS

How much of Equations of Planes have you understood?