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Three planes can intersect along a common line
Three planes can intersect along a common line
The most interesting case is when there is more than one solution.
There are three planes, and they all meet along a common line, like the pages meeting at the spine of a book.
Our calculators can't find any of the solutions, and give the same "Ma ERROR" that appears when there is no solution.
That's okay, because in either case, we need to check to see whether the system is inconsistent, and if it's not, will be close to describing the line.
If the system is consistent, but does not have a unique solution, we will give equations that describe the solution.
Example
Example
Three planes are given by the equations
(eq1) x - 2y + 0.1z = 4
(eq2) 3x - y + 0.5z = 5
(eq3) 2x + y + 0.4z = 1
As usual, we eliminate x from (eq1) and (eq2):
(eq2) - 3(eq1)
5y + 0.2z = -7 (eq4)
Now, eliminate x from (eq1) and (eq3):
(eq3) - 2(eq1)
5y + 0.2z = -7 (eq5)
These are the same equations, and we can see that (eq5) - (eq4) gives
0 = 0.
This tells us that the system is consistent, but we don't have enough information to give a unique solution.
Suppose that z is some real number t . Then
5y + 0.2t = -7
y = -0.04t - 1.4
Working back with (eq3) we find
2x + y +0.4t = 1
2x - 0.04t - 1.4 + 0.4t = 1
2x = 2.4 -0.36t
x = 1.2 - 0.18t
So a solution is in the form
(x,y,z) = (1.2 - 0.18t, -0.04t - 1.4, t ) where t is any real number.
You could check by substituting back into the original three equations that this is a point on each plane.
You could find some particular points on the line by substituting in values of t :
at t=0, (x,y,z) = (1.2, -1.4, 0)
at t=1, (x,y,z) = (1.02, -1.44, 1)
at t=2.5, (x,y,z) = (0.75, -1.5, 2.5)
The three planes intersecting along a common line.
A system of equations is consistent if it has solutions.
We find in this situation that there are many solutions, in the form of a line.
From many solutions to none in one small step
From many solutions to none in one small step
Think about what would happen to the planes given above if one of them moved slightly. It stays parallel to the original plane, shifts slightly. For instance, the second planes shifts from
(eq2) 3x - y + 0.5z = 5
to become
(eq2) 3x - y + 0.5z = 2 .
The plane for (eq2) is shifted (while still parallel to the original plane). A triangular prism forms.
We can see that only the purple plane (which represents the points satisfying (eq2)) has moved. Where there was a line of solutions before, now there is no solution.
In fact, if the constant term in 3x - y + 0.5z = D is anything other than D = 5, the system will be inconsistent.
Exercises
Exercises
For each set of equations, find an equation for the line of intersection as a function of a variable t that can take any real value.
A:
x + 2y + 3z = 0
x + y + z = 0
3x + 2y + z = 0
B:
x - 2y - 2z = 0
3x + 2y + z = 2
x + 14y + 12z = 4
C:
2x + 0.5y + z = 2
2x + 2y + 2.4z = 3
1.4x + 1.1y + 1.4z = 1.9
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