# The intersection of three-dimensional hyperplanes

• Nov 22nd 2013, 10:48 PM
student2011
The intersection of three-dimensional hyperplanes
Dear mathematicians

I have a question regarding the intersection of three dimensional hyperplanes in R^4. I know that the intersection of two planes in R^2 is a line. So the intersection of a finite number of planes is a line and we can thought of the planes as a pages of an open book.

How can we define the intersection of three-dim hyperplanes in R^4. Is it true that the intersection of finite number of hyperplanes is a common plane between them, if so how can we visualise and sketch this.

Thank you in advance
• Nov 23rd 2013, 12:36 AM
chiro
Re: The intersection of three-dimensional hyperplanes
Hey student2011.

You need to setup a matrix system (Ax=b) corresponding to the hyper-planes and then look at the rank of the solution. That will tell whether it even has a solution and if so whether its a point/line/plane/etc.
• Nov 23rd 2013, 07:18 PM
student2011
Re: The intersection of three-dimensional hyperplanes
Thank you so much chiro, so the intersection of finite number of hyperplanes in R^4 is not necessarily a plane.
• Dec 3rd 2013, 04:41 AM
student2011
Re: The intersection of three-dimensional hyperplanes
Dear assistants

I have another question:

Is it true that " there is n 3-dim hyperlanes in R^4 such that the intersection between them is a plane, for all positive integer n"

Thank you in advance
• Dec 3rd 2013, 08:59 AM
HallsofIvy
Re: The intersection of three-dimensional hyperplanes
Quote:

Originally Posted by student2011
Dear mathematicians

I have a question regarding the intersection of three dimensional hyperplanes in R^4. I know that the intersection of two planes in R^2 is a line.

Did you mean "two planes in R^3"? There cannot be two distinct planes in R^2.
Even assuming you mean R^3, this, itself, is not generally true. The intersection of two planes in R^2 is (1) a line or (2) the empty set or (3) a plane.

Quote:

So the intersection of a finite number of planes is a line and we can thought of the planes as a pages of an open book.

How can we define the intersection of three-dim hyperplanes in R^4. Is it true that the intersection of finite number of hyperplanes is a common plane between them, if so how can we visualise and sketch this.

Thank you in advance
• Dec 5th 2013, 10:07 AM
student2011
Re: The intersection of three-dimensional hyperplanes
Yes, you are right Hallsoflvy. I mean the intersection of planes in R^3 not R^2 there is only one plane in R^2. The intersection is either empty or a plane if they are identical, these two cases are trivial, or a line.

Thanks alot Hallsoflvy