Intersection of Planes

This page is all about how planes meet (or not). Two planes will either be parallel or meet at a line. Three planes will either 1) not meet - zero solutions, 2) meet at a point - unique solution, or 3) meet at a line (or plane) - infinite solutions. We will look at how to recognise each situation and how to find the solutions. This work overlaps with finding solutions to systems of linear equations.


Key Concepts

On this page, you should learn about

  • the intersection of two planes
  • the intersection of three planes (unique, infinite and zero solutions)

Essentials

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

Unique Solution

In this video, we see that systems of equations with 3 unknowns can be represented in 3D space by planes. We use the method of elimination to solve the following system and find the intersection of the 3 planes.

x + y + 2z = 0

2x - y + z = -6

3x + 4y - z = -6

Notes from the video

Zero or Infinite Solutions

Previously we have seen what happens when there is a unique solution to a system of equations. Here we will look at what happens when we do not get a unique solution. In other words, we will consider when there is no solution (zero) and when there are infinite solutions. As well as solving the system of equations algebraically using the elimination method, we will look at what happens graphically: how the three planes that meet (or not).

The two systems of equations that we will use to illustrate this are

x + 3y - 2z = 7

2x - 2y + z = 3

3x + y - z = 12

and

x + 3y - 2z = 7

2x - 2y + z = 3

3x + y - z = 10

Notes from the video

2 Planes

When we consider 2 planes, they can either

  1. be parallel (an therefore never meet)
  2. meet at a line
  3. be coincident

In the following video, we are going to look at the second case and see how to find the equation of the line where two planes meet.The method that we will use here gives the solution just like your calculator would.

We will illustrate this by solving the following system of equations:

x + 3y - 2z = 7

2x - 2y + z = 3

Notes from the video

3 Planes - Infinite Solutions

In the following example, we are going to solve a system of equations. This system has infinite solutions and we will find the general solution. We will see that the 3 equations represent 3 planes and the planes meet at a line. The method is the same as finding the intersection of 2 planes.

Show that the following system of equations has infinite solutions and find the general solution of this system

x + 3y - 2z = 7

2x - 2y + z = 3

3x + y - z = 10

Notes from the video

Problem Solving

This is a typical exam-style question. It requires you to have a good understanding of what happens when you try to solve a system of equations when there is infinite solutions and when there is a unique solution.

When we eliminate one variable…

  • Unique Solution - it works, we can solve it!
  • Zero Solution - 2 inconsistent equations
  • Infinite solutions - 2 identical equations

Find the value(s) of k for which the system of equations below has

  1. Infinitely many solutions
  2. A unique solution

2x - y + z = 0

3x + 2y - z = 8

x + ky + 3z = -k² + 8

Notes from the video

Summary

Print from here

Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

Exam-style Questions

Question 1

Find the intersection of the planes \({ \Pi }_{ 1 } \) and \({ \Pi }_{ 2 }\) in the form \(\textbf {r}=\textbf {a}+\lambda\textbf {b}\) where the components of b are integers

\({ \Pi }_{ 1 }:\quad \quad x+2y−z=5\\ { \Pi }_{ 2 }:\quad \quad 2x−y+3z=−4\)

Hint

Full Solution

 

Question 2

\(\ \ \ x + \ \ y +\ \ z =8\\ \ \ ax − \ y \ \ \quad \ \ \ =3\\ −x+3y+4z=b\)

  1. There is no unique solution solution to the system of equations. Find the value of a.
  2. Given that the system can be solved, find the value of b.

Hint

Full Solution

 

Question 3

The three planes \({ \Pi }_{ 1 }\) , \({ \Pi }_{ 2 }\) and \({ \Pi }_{ 3 }\) meet at a line

\({ \Pi }_{ 1 }:\quad 2x+\ y+3z=a\\ { \Pi }_{ 2 }:\quad \ x−2y+2z=−9 \\ { \Pi }_{ 3 }:\quad 3x+4y+4z=−1\)

  1. Find a
  2. Find the equation of the straight line in the form \(\textbf {r}=\textbf {a}+\lambda\textbf {b}\) where the components of b are integers

Hint

Full Solution

 

Question 4

Find the value of k which makes the following system of equations inconsistent:

\(x +2y+kz=−1\\ 2x+ \ y− \ z=3\\ kx−2y+ z=1\)

Hint

Full Solution

 

MY PROGRESS

How much of Intersection of Planes have you understood?