# How Many Movable Ways(vector) In Pure $N$th-Dimensional Space?

In my opinion, In pure $2$th-dimensional space, There is 2 movable ways.

And in pure $3$th-dimensional space, There is 3 movable ways.

Am I think in right way?

Any answers will be appreciated, thank you.

Added. Here is what I'm thinking about "Real Space"

Two dots are particles.(such as electron)

The lines which they release is quantized.

So it can be measurable or comparable distance between two dots.

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What is a "movable way"? – Qiaochu Yuan Jul 17 '11 at 21:21
It can move in infinitely many ways. Can you be more precise about what you're talking about? I don't know an interpretation of this question that isn't a tautology. – Qiaochu Yuan Jul 17 '11 at 21:30
@Qiaochu: I think his question is "how large is the basis in an $N$ dimensional vector space," but I am not sure. – Eric Naslund Jul 17 '11 at 21:38
@Qiaochu : What he is trying to ask is that how many degrees of freedom are there in n-dimensional space, which I think the answer is the sum of Permutations(1,n) + Permutations(2,n) + ...+Permutations(n-1,n) degrees of freedom. – Arjang Jul 18 '11 at 11:47
See also How Many Movable Ways(Direction) In Pure Nth-Dimensional Space?, posted on Physics by same user. – Dori Jul 18 '11 at 21:59

## 1 Answer

In $N$ dimensional space there are $N$ basis vectors, and hence $N$ "directions".

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@kso830: What do you mean by real space? A real vector space? – Eric Naslund Jul 17 '11 at 21:29
@kso830: I guess that depends. The Minkowski spacetime is 4-dimensional, as there are 3 spacial dimensions and 1 dimension of time. Although I am to understand that in string theory (which I know nothing about) there are more spacial dimensions which appear as compact manifolds which we cannot see. It might be best to ask that on the physics forum. – Eric Naslund Jul 17 '11 at 21:37
For purposes of this question, I would just say we live in a 3-dimensional space and leave it at that. The possibility of extra dimensions is an open question but it's pretty clear that they have no effect on anything at normal length scales. – David Z Jul 18 '11 at 7:27
@David Zaslavsky: How can you guarantee that "they have no effect on anything at normal length scales"?? – Xiang Jul 18 '11 at 8:01
@David: "Just look around, do you see any evidence of extra dimensions?" Really? This last line is some of the worst reasoning possible, really some of the absolute worst. Look around, do you see any evidence the world is a sphere? Look around, do you see any evidence the sun is gigantic? You can apply that logic to anything, and arrive at ridiculousness. In short, your intuition (and mine) about the world is wrong, and you should not trust it. A better example: Look around, does space seem to be curved? No, but it is, and without factoring in GR, GPS devices will not work. – Eric Naslund Jul 18 '11 at 14:57