Файл:Hesse normalenform.svgDistance from the origin O to the line E calculated with the Hesse normal form. Normal vector in red, line in green, point O shown in blue.
The dot <math>\cdot</math> indicates the scalar product or dot product.
Vector <math>\vec r</math> points from the origin of the coordinate system, O, to any point P that lies precisely in plane or on line E. The vector <math>\vec n_0</math> represents the unitnormal vector of plane or line E. The distance <math>d \ge 0</math> is the shortest distance from the origin O to the plane or line.
Derivation/Calculation from the normal form
Note: For simplicity, the following derivation discusses the 3D case. However, it is also applicable in 2D.
In the normal form,
<math>(\vec r -\vec a)\cdot \vec n = 0\,</math>
a plane is given by a normal vector <math>\vec n</math> as well as an arbitrary position vector <math>\vec a</math> of a point <math>A \in E</math>. The direction of <math>\vec n</math> is chosen to satisfy the following inequality
<math>\vec a\cdot \vec n \geq 0\,</math>
By dividing the normal vector <math>\vec n</math> by its magnitude <math>| \vec n |</math>, we obtain the unit (or normalized) normal vector
<math>\vec n_0 = {{\vec n} \over {| \vec n |}}\,</math>
and the above equation can be rewritten as
<math>(\vec r -\vec a)\cdot \vec n_0 = 0.\,</math>
In this diagram, d is the distance from the origin. Because <math>\vec r \cdot \vec n_0 = d</math> holds for every point in the plane, it is also true at point Q (the point where the vector from the origin meets the plane E), with <math>\vec r = \vec r_s</math>, per the definition of the Scalar product
