Thread: How 3 planes intersect.

Hey, thanks for coming into my thread!

I have a problem. I am given 3 equations of planes, in the form Ax + By + Cz + D = 0, and am being asked how they are intersecting.

Here are the 3 equations:
1. x + 2y + 3z + 4 = 0
2. x - y - 3z - 2 = 0
3. x + 5y + 9z + 10 = 0

I decided to run it through Gauss Jordan elimination to see if they intersect at the same point:
***I am having trouble working with matrices with latex, so hopefully this makes sense***

Starting with:
1 2 3 -4
1 -1 -3 2
1 5 9 -10

So I subtract line 1 from 2 and line 1 from 3:
1 2 3 -4
0 -3 -6 6
0 3 6 -6

Then I multiply line 1 by 2 and line 2 by -1:
2 4 6 -8
0 3 6 -6
0 3 6 -6

Then I subtract line 2 from line one, and line 2 from line 3 which leaves:
2 1 0 -2
0 3 6 -6
0 0 0 0

I have not encountered this before using this technique. What does it mean? Also, what does it mean with regards to how the planes intersect?

Thank you! Please feel free to ask questions, I will gladly clear anything up that I said here.

EDIT: It was a typing mistake. I meant to put -1 not -2. Which makes this make more sense!

Originally Posted by Kakariki

Hey, thanks for coming into my thread!

I have a problem. I am given 3 equations of planes, in the form Ax + By + Cz + D = 0, and am being asked how they are intersecting.

Here are the 3 equations:
1. x + 2y + 3z + 4 = 0
2. x - y - 3z - 2 = 0
3. x + 5y + 9z + 10 = 0

I decided to run it through Gauss Jordan elimination to see if they intersect at the same point:
***I am having trouble working with matrices with latex, so hopefully this makes sense***

Starting with:
1 2 3 -4
1 -2 -3 2
1 5 9 -10

So I subtract line 1 from 2 and line 1 from 3:
1 2 3 -4
0 -3 -6 6 This line is incorrect, it should be 0 -4 -6 6. You will need to recompute everything else from here.
0 3 6 -6

Then I multiply line 1 by 2 and line 2 by -1:
2 4 6 -8
0 3 6 -6
0 3 6 -6

Then I subtract line 2 from line one, and line 2 from line 3 which leaves:
2 1 0 -2
0 3 6 -6
0 0 0 0

I have not encountered this before using this technique. What does it mean? Also, what does it mean with regards to how the planes intersect?

Thank you! Please feel free to ask questions, I will gladly clear anything up that I said here.

See my comment in the quote...

It means that equation three is a linear combination of equation's one and two.

Originally Posted by Kakariki

Hey, thanks for coming into my thread!

I have a problem. I am given 3 equations of planes, in the form Ax + By + Cz + D = 0, and am being asked how they are intersecting.

Here are the 3 equations:
1. x + 2y + 3z + 4 = 0
2. x - y - 3z - 2 = 0
3. x + 5y + 9z + 10 = 0

I decided to run it through Gauss Jordan elimination to see if they intersect at the same point:
***I am having trouble working with matrices with latex, so hopefully this makes sense***

Starting with:
1 2 3 -4
1 -2 -3 2
1 5 9 -10

So I subtract line 1 from 2 and line 1 from 3:
1 2 3 -4
0 -3 -6 6
0 3 6 -6

Then I multiply line 1 by 2 and line 2 by -1:
2 4 6 -8
0 3 6 -6
0 3 6 -6

Then I subtract line 2 from line one, and line 2 from line 3 which leaves:
2 1 0 -2
0 3 6 -6
0 0 0 0

If this were in fact correct, you could divide the second line by 3 to get
2 1 0 -2
0 1 2 -2
0 0 0 0
then subtract the second line from the first to get
2 0 -2 0
0 1 2 -2
0 0 0 0

which corresponds to the equations 2x- 2z= 0 and y+ 2z= -2. That would mean that you can write both x and y in terms of z: x= z and y= -2- 2z. If take z itself to be a parameter, that's the equation of a line: x= t, y= -2- 2t, z= t. Here the three planes intersect in a single line.

But, as Prove It said, you have made an arithmetic error. In fact, these three planes do intersect in a single point.

Originally Posted by HallsofIvy

If this were in fact correct, you could divide the second line by 3 to get
2 1 0 -2
0 1 2 -2
0 0 0 0
then subtract the second line from the first to get
2 0 -2 0
0 1 2 -2
0 0 0 0

which corresponds to the equations 2x- 2z= 0 and y+ 2z= -2. That would mean that you can write both x and y in terms of z: x= z and y= -2- 2z. If take z itself to be a parameter, that's the equation of a line: x= t, y= -2- 2t, z= t. Here the three planes intersect in a single line.

But, as Prove It said, you have made an arithmetic error. In fact, these three planes do intersect in a single point.

This is exactly what I wanted to hear! I was given diagrams of graphs to match this to, and the one I chose was the one that had the 3 planes intersecting in a line.

Thank you all for your help!!! Sorry for the typo in my original post, proves you are all very keen in your arithmetic!