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Any 3 points define a plane and therefore ALWAYS lie on the same plane. The same goes for two points and a straight line.

Does this mean that a hyperplane in four dimensional space is defined by any four points? Can the concept be applied to higher dimensions and still make sense or be useful?

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    $\begingroup$ Three points determine a plane if the points are not all on the same line. Four points determine a 3-dimensional hyperplane if the points are not all on the same plane. $\endgroup$ – Doug Chatham Jul 29 '16 at 13:29
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    $\begingroup$ If you know some linear algebra then you'll know that a hyperplane in $\Bbb R^n$ is an $(n-1)$-dimensional subspace; i.e. it is the span of $n-1$ linearly independent vectors. Given a set of $n$ distinct points $\{p_1, \dots, p_n\}$ you can always create $n-1$ vectors by taking one as the base point: $\{\overrightarrow{p_1p_2},\overrightarrow{p_1p_3},\dots,\overrightarrow{p_1p_n}\}$. Then those $n-1$ vectors will span an $(n-1)$-dimensional space if they are all linearly independent (equivalently if the points are affinely independent). $\endgroup$ – user137731 Jul 29 '16 at 13:30
  • $\begingroup$ A hyperplane is an $(n-1)$-dimensional affine subspace, not necessarily a linear subspace. $\endgroup$ – GEdgar Jul 29 '16 at 15:43
  • $\begingroup$ What is your mathematical background? $\endgroup$ – Daan Michiels Jul 29 '16 at 15:50
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    $\begingroup$ I did, but I don't know what HL IB maths means :) $\endgroup$ – Daan Michiels Jul 29 '16 at 15:53
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Does this mean that a hyperplane in four dimensional space is defined by any four points?

Yes, if they are not all in the same 2-dimensional plane. (The same way that 2 point determine a line if they're not equal, and 3 points determine a 2-dimensional plane if they're not all in a line.)

In general, in a space of dimension at least $n$: if you pick $n+1$ points in the space, they will determine a unique "plane" of dimension $n$, unless they're all in the same "plane" of dimension $n-1$.

Can the concept be applied to higher dimensions and still make sense or be useful?

Yes. These things are typically taught in a course in Linear algebra. In linear algebraic terms, an $n$-dimensional "plane", as I was referring to it, is called an $n$-dimensional affine subspace.

Linear algebra is arguably one of the most fundamental pieces of math in use today, so it should certainly be considered useful.

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