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2 points make a line -1D

3 points at most make a plane -2D

similary can we find $n$ points making up a $n-1$-dimensional object in $\mathbb{R}^n$?

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Yes, in some sense. Try reading something about "affine spaces" and their subspaces. – Giuseppe Negro Jan 3 '13 at 19:13
@GiuseppeNegro: What 4 points would make? I mean, a general name. – Inquisitive Jan 3 '13 at 19:23
If they are in general position they span a 3-dimensional space. – Giuseppe Negro Jan 3 '13 at 19:25
up vote 1 down vote accepted

Yes, under certain conditions. Let $p,q$ be points. Then denote by $v=p-q$ the vector from $q$ to $p$.

Let $p_0,...,p_{n-1}$ be $n$ point in $n$-space, make the $n-1$ vectors $v_1=p_1-p_0,...,v_{n-1}=p_{n-1}-p_0$. If this set of vectors is linearly independent, then the points uniquely define an $n-1$-dimensional hyperplane in $n$-space.

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