Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Equation of a Plane in R^n , n>3

  1. Sep 2, 2011 #1


    User Avatar
    Science Advisor
    Gold Member


    Just curious: what is the equation of a plane in R^n for n>3 ?

    We cannot define a normal vector for n>3 , so, what do we do? I thought

    of working with flat, embedded copies of R^2 . In terms of linear algebra,

    I guess we could see a plane in R^n as any image of a linear map with

    rank=2. In terms of geometry, maybe we need a "flat" embedding of R^2

    (I am not clear how to make the term 'flat' here more precise) in R^n.

    Anyone know?

  2. jcsd
  3. Sep 2, 2011 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    In terms of [itex]\mathbb{R}^n[/itex], then for [itex]\boldsymbol{x} = [x_1,x_2,\ldots,x_n]^\text{T}\in\mathbb{R}^n[/itex] and non-zero scalars [itex]a_n[/itex] the sub-space

    [tex]\text{const.} = \sum_{i=1}^n a_ix_i[/tex]

    is a hyperplane of [itex]\mathbb{R}^n[/itex]. In actuality, the definition of a hyperplane is more compact: A hyperplane of any vector space is any vector subspace of co-dimension 1.
  4. Sep 2, 2011 #3


    User Avatar
    Science Advisor
    Gold Member

    Thganks, but I was thinking of a 2-d plane living in R^n with n higher than 2.

    would that still be defined as a1.x1+a2.x2+a3.x3+0x4+...+0.xn=constant?
  5. Sep 2, 2011 #4


    User Avatar
    Science Advisor

    No. It cannot be done with a single equation like that. To identify an m-dimensional object in n-dimensional space requires n- m numerical equations. That is why Hootenanny was able to give a single equation for a hyper-plane (codimension 1 so dimension n- 1). To determine a 2 dimensional plane in n dimensional space would require n- 2 numerical equations.
  6. Sep 3, 2011 #5
    It would have n-2 linearly independent normal vectors. Take the intersection of the hyperplanes passing through a given point, each with one of the normal vectors.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Similar Threads for Equation Plane
I Deduce Geodesics equation from Euler equations