Is a hyperplane defined by four points?

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?

• 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. – Doug Chatham Jul 29 '16 at 13:29
• 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). – user137731 Jul 29 '16 at 13:30
• A hyperplane is an $(n-1)$-dimensional affine subspace, not necessarily a linear subspace. – GEdgar Jul 29 '16 at 15:43
• What is your mathematical background? – Daan Michiels Jul 29 '16 at 15:50
• I did, but I don't know what HL IB maths means :) – Daan Michiels Jul 29 '16 at 15:53

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 $n-1$.
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.