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Math Help - The intersection of three-dimensional hyperplanes

  1. #1
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    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
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  2. #2
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    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.
    Thanks from Shakarri and student2011
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  3. #3
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    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.
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  4. #4
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    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
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  5. #5
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    Re: The intersection of three-dimensional hyperplanes

    Quote Originally Posted by student2011 View Post
    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.

    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
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  6. #6
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    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
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