Define: Hyperplane

Define: Hyperplane

In geometry a hyperplane is a subspace of one dimension less than its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines.

--wikipedia

A hyperplane in n$n$-dimensional space is an (n−1)$(n-1)$-dimensional object that can be described by n⃗ ⋅x⃗ =k$\overrightarrow{n}\cdot \overrightarrow{x}=k$ where n⃗ $\overrightarrow{n}$ is a constant vector orthogonal to the hyperplane, x⃗ $\overrightarrow{x}$ is a variable vector from the origin to a point on the plane, and k$k$ is some scalar constant. A hyperplane in 2-space is a line; a hyperplane in 3-space is a plane. (edit: Matt E.'s comment that a hyperplane is a subspace of dimension one less than the whole space is a much nicer definition than mine.)

--By Isaac from stackexchange