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.
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
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:
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
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
The following system of equations is inconsistent. Fill in the blanks and find the value of k
\(\ x + 3y - 2z = k² + 1\\ 2x- 2y + \ z = 4\\ kx + \ y - \ z = 6\)
A: x + 3y - 2z = k² + 1
B: 2x - 2y + z = 4
C: kx + y - z = 6
B+ C: (2+k)x - y =
A + 2B: 5x - y = k² +
The equations are inconsistent when k =
B+ C: (2+k)x - y = 10
A + 2B: 5x - y = k² + 9
(2 + k) = 5
k = 3
If we substitute this value into the equations:
B+ C: 5x - y = 10
A + 2B: 5x - y = 18
This is not possible, so the equations are inconsistent when k = 3
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