# 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.

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

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