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Let $c_{1}, c_{2}$ be nonzero real numbers and $U$ be the line of equations $c_{1}x_{1}=c_{2}x_{2}$. Find (describe by an equation, equations or in terms of x and y) the orthogonal complement of $U$ with respect to the standard dot product. What does this complement represent geometrically?

EDIT: added a full problem similar to the one I am working on for context.

I am especially confused by the wording "line of equations". Does this mean that the equation describes a line and U is all points on that line?

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  • $\begingroup$ By "translate this to set-builder notation," do you mean "write the solution set in set-builder notation"? If so, the answer is $\{(x_1,x_2):c_1x_1=c_2x_2\}$. I am not sure I understand the question in your title or its relevance to the question in the body. $\endgroup$ – arctic tern May 1 '17 at 21:21
  • $\begingroup$ Well, I assume $U$ must be a set or vector space of some sort, which I am used to seeing in set builder notation, not "U is the line of equations". I was asking how to take it from a "line of equations" to what I have seen more often, which is set builder notation. $\endgroup$ – rocksNwaves May 1 '17 at 22:00
  • $\begingroup$ Perhaps instead of bits and pieces you can provide the original context in full. $\endgroup$ – arctic tern May 1 '17 at 22:20
  • $\begingroup$ You can't really put equations themselves in set-builder notation. Could you give an example of what it is you're looking for? $\endgroup$ – Blavius May 1 '17 at 22:21
  • $\begingroup$ @arctictern I included a full problem with different equations and dimension than the one I am working on. It's for an assignment, so I was trying to be a little vague in order to make sure I do the brunt of the work myself. $\endgroup$ – rocksNwaves May 1 '17 at 22:36

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