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.
On this page, you should learn about
- the intersection of two planes
- the intersection of three planes (unique, infinite and zero solutions)
The following videos will help you understand all the concepts from this page
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Here is a quiz that practises the skills from this page
START QUIZ!
Each of the following graphs represents intersecting planes with either a unique, zero or an infinite number of solutions.
Drag the correct answer into the space provided.
unique zero infinite
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Each of the following systems has a unique, zero or an infinite number of solutions.
Drag the correct answer into the space provided.
zero infinite unique
\(x -2y=5\\ 2x-4y=7\)
\(x + y=6\\ 3x+3y=18\)
\(x + 2y=5\\ 2x+y=4\)
\(2x -3y=5\\ -6x+9y=15\)
We can re-write each of the systems and it becomes clear
\(2x -4y=10\\ 2x-4y=7 \) equations are inconsistent => zero solutions
\(3x + 3y=18\\ 3x+3y=18\) equations are identital => infinite solutions
\(x + 2y=5\\ 2x+y=4\) equations are not parallel => unique solution
\(-6x+9y=-15\\ -6x+9y=15\) equations are inconsistent => zero solutions
The point of intersection of the following three planes is (a,b,c). Find the coordinates using your calculator,
\(\ \ x+2y-\ z=5\\ 2x-\ y+3z=-4\\ \ \ x-\ y-\ \ z=2\\\)
a =
b =
c =
Use the simulataneous equation solver on your calculator
Find the intersection of the plane x+2y-3z=8 with the OXY plane
The OXY plane is the plane where z=0
x+2y-3z=8
x+2y-3(0)=8
x+2y=8
Three planes intersect at a point
\(2x+3y-\ z=0\\ \\ x-\ \ y-\ \ z=2\\ 3x-2y+2z=10\)
Fill in the blanks to complete the solution below
A: 2x + 3y - z = 0
B: x - y - z = 2
C: 3x - 2y + 2z = 10
A-B: x + 4 y =
2A+C: 7x + 4y =
6x =
x =
y =
z =
x + 4y = -2
7x + 4y = 10
6x = 12
x = 2
Substitute into
x + 4y = -2
2 + 4y = -2
y = -1
Substitute into
x - y - z = 2
2 - (-1) - z = 2
z = 1
The following system of equations has infinite solutions. Find a and b.
\(\textbf{a}x+3y=\textbf{b}\\ 4x+6y=30\)
a =
b =
\((\textbf{2}x+3y=\textbf{15} )\times2\\ 4x+6y=30\)
The following system of equations has no solution. Find a.
\(15x-5y=8\\ \textbf{a}x+2y=5\)
a =
\((15x-5y=8)\times2 \ \ \ \quad => \quad 30x-10y=16\\ (\textbf{-6}x+2y=5)\times-5\quad => \quad 30x-10y=-25\\ \) equations are inconsistent => zero solutions
The following system of equations has no solution. Find k given that k<0
\(2kx-9y=10\\ 8x-ky=7\)
k =
\((2kx-9y=10)\times4\quad => 8kx-36y=40\\ (8x-ky=7)\times k \quad \quad =>8kx-k²y=7k\\ k²=36\\ k=-6 (k<0) \)
The following system of equations has infinite solutions. Find a and b given that a>0
\(2x-(a+1)y=b\\ ax-\quad \quad10y =30\)
a =
b =
\((2x-(a+1)y=b)\times a \ \ \quad => 2ax-a(a+1)y=ba \\ (ax-\quad \quad10y =30)\times 2 \quad => 2ax- \quad \quad 20y=60\\ a(a+1)=20\\ a^2+a-20=0\\ (a+5)(a-4)=0\\ a=4 (a>0)\)
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
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
\(\ \ \ x + \ \ y +\ \ z =8\\ \ \ ax − \ y \ \ \quad \ \ \ =3\\ −x+3y+4z=b\)
- There is no unique solution solution to the system of equations. Find the value of a.
- Given that the system can be solved, find the value of b.
Hint
Full Solution
The three planes \({ \Pi }_{ 1 }\) , \({ \Pi }_{ 2 }\) and \({ \Pi }_{ 3 }\) meet at a line
\({ \Pi }_{ 1 }:\quad 2x+\ y+3z=a\\ { \Pi }_{ 2 }:\quad \ x−2y+2z=−9 \\ { \Pi }_{ 3 }:\quad 3x+4y+4z=−1\)
- Find a
- 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
How much of Intersection of Planes have you understood?
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