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When we talk about the dimension of a quantity (e.g. the dimension of acceleration is$[ L \ T ^ {-2}]$) are we talking about the same "dimension" as when we talk about three dimensional space?

Are these two separate definitions sharing a word? or are they two uses of the same meaning?

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The dimension of a space is the dimensional analysis dimension of volumes, when a unit of length is given. – Ron Maimon May 16 '12 at 5:53
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The most straightforward, nontechnical explanation is that these are two different meanings for the same word.

However, you can actually think of the units of a quantity as an element in a vector space, whose dimensions are labeled $M$, $L$, $T$, etc., and you can also think of regular three dimensional space as a vector space, whose dimensions are labeled $x$, $y$, and $z$. With this, it's still not quite the same meaning of "dimension," since you might say $L T^{-2}$ is a dimension but you wouldn't say $y - 2z$ is a dimension; however, that does help motivate why the same word is used for both concepts.

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Thank you for your answer! I think that "vector space" was what I needed to hear. – MatrixManAtYrService May 16 '12 at 17:17

Not exactly. Yes, the "dimension" in dimensional analysis refers to the powers, and the "dimension" in "spatial dimension" refers to the directions. In this sense, it is two meanings for one word.

(I'm using "degree of freedom" in a loose sense here)

But, if you look at the more basic meaning of the word--"a 'degree of freedom'", then, both fall within this meaning. Measurable systems are many times classified by saying that the system is "N dimensional", where N refers to the number of variables present. For a point object at rest in space, we have 3 spatial DOFs--so the universe is "3 dimensional" (time can be added to the mix easily, and, less easily, you can talk about adding more spatial dimensions in String Theory).

Similarly, mass, length, time, current, etc can also be used to classify data, giving rise to similar set of degrees of freedom.

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