| # Hyperplanes Vectors > Basics | Scalar product | Norms | Projection on a line | Orthogonalization | Hyperplanes | Linear functions | Application Hyperplanes Projection on a hyperplane Geometry Half-spaces ## Hyperplanes A hyperplane is a set described by a single scalar product equality. Precisely, an hyperplane in is a set of the form where , , and are given. When , the hyperplane is simply the set of points that are orthogonal to ; when , the hyperplane is a translation, along direction , of that set. If , then for any other element , we have Hence, the hyperplane can be characterized as the set of vectors such that is orthogonal to : Hyperplanes are affine sets, of dimension (see the proof here). Thus, they generalize the usual notion of a plane in . Hyperplanes are very useful because they allows to separate the whole space in two regions. The notion of half-space formalizes this. Example: A hyperplane in . ## Projection on a hyperplane Consider the hyperplane , and assume without loss of generality that is normalized (). We can represent as the set of points such that is orthogonal to , where is any vector in , that is, such that . One such vector is . By construction, is the projection of on . That is, it is the point on closest to the origin, as it solves the projection problem Indeed, for any , using the Cauchy-Schwartz inequality: and the minimum length is attained with . ## Geometry of hyperplanes
| Geometrically, an hyperplane , with , is a translation of the set of vectors orthogonal to . The direction of the translation is determined by , and the amount by . Precisely, is the length of the closest point on from the origin, and the sign of determines if is away from the origin along the direction or . As we increase the magnitude of , the hyperplane is shifting further away along , depending on the sign of . In the image on the left, the scalar is positive, as and point to the same direction.
## Half-spaces A half-space is a subset of defined by a single inequality involving a scalar product. Precisely, an half-space in is a set of the form where , , and are given. Geometrically, the half-space above is the set of points such that , that is, the angle between and is acute (in ). Here is the point closest to the origin on the hyperplane defined by the equality . (When is normalized, as in the picture, .)
| The half-space is the set of points such that forms an acute angle with , where is the projection of the origin on the boundary of the half-space.