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I was reading up a little on reference frames and am confused about a certain detail. Is it correct to say that the positive $x$-axis, for example, can be defined as follows?

$$ X^+ = \lim_{C \rightarrow \, 0^+} C^{-1}\langle 1,0,0\rangle \ \ \ \ \ \ \ \ \ \ \ \ (1) $$

Here, $\langle a_1, a_2, a_3\rangle$ represents the vector with origin at $(0,0,0)$ and terminal point at $(a_1, a_2, a_3)$. Also, this is just a question I have, not something I read. Thanks to all in advance.

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There was an error concerning $C$. I've fixed that now. – ThisIsNotAnId Mar 7 '12 at 0:42
up vote 2 down vote accepted

No, it wouldn't be correct to say that. Firstly, the limit of a sequence of vectors is just another vector (if it exists); and secondly, that sequence of vectors has no limit. Also, I'd say that an axis consists of points, not vectors.

So a correct alternative would be: the set $$\{(a_1,0,0)\in\mathbb{R}^3\mid a_1>0\}$$ forms the positive $x$-axis in $\mathbb{R}^3$. You could also use $$\displaystyle\bigcup_{C>0}\{C(1,0,0)\}.$$

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Oh, yeah! I knew I was missing something. Thanks! – ThisIsNotAnId Mar 7 '12 at 2:17
And, btw, really nice set of quotes on your site. – ThisIsNotAnId Mar 7 '12 at 2:22
Thank you, I enjoy rereading them often :) – Zev Chonoles Mar 7 '12 at 3:57

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