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werdna91

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## Homework Statement

Here is a picture of the problem:

http://img84.imageshack.us/img84/1845/screenshot20100927at111.png

If the link does not work, the problem basically asks:

Let "a" be a non-zero vector in R^n. Let S be the set of all orthogonal vectors to "a" in R^n. I.e., a•x = 0 (where • denotes dot product).

Prove that the interior of S is empty.

## Homework Equations

B(r, a) = { x in R^n : |x - a| < r }

## The Attempt at a Solution

I realize that I basically have to show that S is its own boundary. In other words, given a vector x in S, I must show that for every open ball B(r, s) (centred at s with radius r > 0), it contains points that are in S and points that are in the complement of S.

My trouble lies with somehow linking dot product with distance .

So if I say, let x be in S, and let y be in S^c (the complement of S).

How can I show that for all r > 0, y is in B(r, x)? All I know is that a•y ≠ 0. Clearly the norm of the vectors are irrelevant to their existence in the set S. But norm obviously has something to do with their existence in B(r, x).

Perhaps I'm over-thinking this. Clarification and suggestions for next steps would be great.

Thanks a lot.

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