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Homework Help: Interior and boundary of set of orthogonal vectors

  1. Oct 3, 2012 #1
    Let "a" be a non zero vector in R^n and define S = { x in R^n s.t. "a" · "x" = 0}. Determine S^int , bkundary of S, and closure of S. Prove your answer is correct


    Ok im more sk having trouble proving that the respective points belong to its condition. Such as thr interior. I know will be all points in thr plane of X, but how can i show that all thise points are the interior if S? All i can think of is letting a point, call it "y", be in thr B(r,a) and showing that it is a interior point through the triangle inequality manipulation. But that doesn't use the fact of orthogonality in the set.
  2. jcsd
  3. Oct 3, 2012 #2


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    If a is a single vector in Rn, then the set of all vectors that are orthogonal to a is an n-1 dimensional "plane". You shouldn't have to use any "neighborhoods" or other technical details.

    When in doubt, look at simple variations of the problem. Suppose n= 2 and a= <1, 0>. The the set of all vectors orthogonal to a is {<0, y>} for any y- the entire y- axis. What are the boundary, interior, and closure of the y- axis? Suppose n= 3 and a= <0, 0, 1>. The set of all vectors orthogonal to a is {<x, y, 0>} for any x and y- the xy-plane. What are the boundary, interior, and closure of the xy-plane?
  4. Oct 3, 2012 #3
    ok, well using the examples you gave. The boundary, interior, and closure for n= 2: y- axis is the boundary, empty interior, and y axis is the points in the closure.

    Now I tried extending it to n-1 imagining a plane and I got a similar result I believe:

    Boundary would be all points in the n-1 plane, interior would be empty, and the closure would be all the boundary points
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