An hyperplane in 3D

Consider an affine set of dimension in $mathbf{R}^3$ , which we describe as the set of points $x in mathbf{R}^3$ such that there exists two parameters lambda_1,lambda_2 such that

$x = left(begin{array}{c} 3lambda_1-4lambda_2 + 4 lambda_1 lambda_2 end{array}right) = left(begin{array}{c} 4 0 0 end{array}right) + lambda_1 left(begin{array}{c} 3 1 0 end{array}right) + lambda_2 left(begin{array}{c} -4 0 1 end{array}right).$

The set mathbf{H} can be represented as a translation of a linear subspace: $mathbf{H} = x_0 + mathbf{L}$ , with

$x_0 := left(begin{array}{c} 4 0 0 end{array}right),$

and mathbf{L} the span of the two independent vectors

u := left(begin{array}{c} 3 1 0 end{array}right) , ;; v := left(begin{array}{c} -4 0 1 end{array}right).

Thus, the set mathbf{H} is of dimension in $mathbf{R}^3$ , hence it is an hyperplane. In $mathbf{R}^3$ , hyperplanes are ordinary planes.

We can find a representation of the hyperplane in the standard form

$mathbf{H} = left{ x ~:~ a^T(x-x_0)=0 right}.$

We simply find that is orthogonal to both and . That is, we solve the equations

0 = a^Tu = 3a_1 +a_2 = 0, ;; 0 = a^Tv = -4a_1 + a_3.

The above leads to a = (a_1, -3a_1, 4a_1) . Choosing for example a_1 = 1 leads to a=(1,-3,4) .

The hyperplane mathbf{H} can be expressed as $x_0+mbox{bf span}(u,v)$ , where x_0 is a particular element, and u,v are two independent vectors. The set is represented in light blue; it is a translation of the corresponding span mathbf{L} =mbox{bf span}(u,v) . Any point x in mathbf{H} is such that x-x_0 belongs to . Thus we can represent the hyperplane as the set of points such that x-x_0 is orthogonal to , where is any vector orthogonal to both u,v .