| # An hyperplane in 3D Consider an affine set of dimension in , which we describe as the set of points such that there exists two parameters such that The set can be represented as a translation of a linear subspace: , with and the span of the two independent vectors Thus, the set is of dimension in , hence it is an hyperplane. In , hyperplanes are ordinary planes. We can find a representation of the hyperplane in the standard form We simply find that is orthogonal to both and . That is, we solve the equations The above leads to . Choosing for example leads to .
| The hyperplane can be expressed as , where is a particular element, and are two independent vectors. The set is represented in light blue; it is a translation of the corresponding span . Any point is such that belongs to . Thus we can represent the hyperplane as the set of points such that is orthogonal to , where is any vector orthogonal to both .