Embedded Files
Sometimes, three planes don't intersect at all
Sometimes, three planes don't intersect at all
The intersection points of three planes are the points that are on all three plane.
We saw in the previous section that there can be a unique point of intersection, like the corner of a room.
However, sometimes there may be no point in common.
Our graphics calculators will give the "Ma ERROR" message when trying to solve systems with no solution. However, they will also give this error message when there are many solutions.
When there is no solution to a system of equations, it will be inconsistent.
Parallel Planes
Parallel Planes
If two planes are parallel (but not the same plane repeated), then there are no points on both planes.
Adding a third plane into the mix, whether parallel to the others or not, will not change the fact that there is no point of intersection.
We already saw in 2: Planes how to tell if two planes are parallel.
Two parallel planes.
Three parallel planes.
Two parallel planes with a third plane not parallel to the first two.
Example
Example
Jane buys four burgers, two drinks and two sundaes for $15.50.
Kyle buys two burgers, one drink and one sundae for $8.50.
Lana buys three burgers and two sundaes for $12.50.
What are the prices of the burgers, drinks and sundaes.
Solution
Solution
Let b, d and s be the prices of burgers, drinks and sundaes respectively.
(eq1) 4b + 2d + 2s = 15.5
(eq2) 2b + d + s = 8.5
(eq3) 3b + 2s = 12.5
Take half of (eq1)
(eq4) 2b + d + s = 7.75
From (eq2) and (eq4) we see
8.5 = 7.75
This is inconsistent; there are no solutions to this system of equations.
The planes are parallel - we can see this because they are now both in the form Ab + Bd + Cs = D .
This means no prices work to fit both (eq1) and (eq2) at the same time.
The purple and red planes represent possible prices that fit the purchases made by Jane and Kyle.
The green plane represents possible prices to fit the purchase made by Lana.
No solution with parallel lines of intersection
No solution with parallel lines of intersection
Another situation is possible, where there are no points in common to three planes.
Think about the rectangular sides of a Toblerone chocolate box. Any pair of sides meet along a line, but those three lines are parallel, and there is no point on all three planes.
In an algebraic context, the system of equations is inconsistent, without parallel planes.
Example
Example
Caleb buys three rides on the Ferris wheel and one ride on the roller coaster for $15.
Dora buys three rides on the roller coaster and one ride on the terror catapult for $35.
They know that terror catapult rides cost nine times as much as Ferris wheel rides.
What are the possible costs?
Solution
Solution
Let x, y and z be the costs of one ride on the Ferris wheel, roller coaster, and terror catapult respectively.
Three equations relating the prices are:
(Caleb) (eq1) 3x + y = 15
(Dora) (eq2) 3y + z = 35
(ratio) (eq3) 9x = z
Taking (eq1) and (eq2), we can eliminate y:
3(eq1) - (eq2)
9x + 3y = 45
-3y - z = -35
(eq4) 9x - z = 10
(eq5) 9x - z = 0
Trying to eliminate x, we find
(eq4) - (eq5)
0 = 10
The equations are inconsistent, there is no solution.
There is no combination of prices that meet these requirements.
We can see one end of the 'Toblerone' triangular prism in the top of the image.
Exercises
Exercises
Take three planes from the list below. Show that the planes do not have any point of intersection.
Describe geometric nature of the planes.
Do this for all 20 combinations of choices of planes.
PLANE A: x + 2y + 3z = 5
PLANE B: 3x + 2y + z = 6
PLANE C: x + y + z = 3
PLANE D: x - z = 7.7
Google Sites
Report abuse