Let's say we have a point-normal representation of a space:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] n \cdot P_{0}P = 0 [/tex] where n is a vector [itex]<a_{1},a_{2}...a_{n}>[/itex] and [itex]P_{0}[/itex] is a point through which the space passes and P is the set of all points contained in the space.

In [itex]ℝ^{2}[/itex], the point-normal representation defines a line.

In [itex]ℝ^{3}[/itex], the point-normal representation defines a plane.

In [itex]ℝ^{n}[/itex], the point-normal representation defines a hyperplane, or (n-1) dimensional affine space.

It can be shown that this affine space can be represented in component form as:

[tex] a_{1}x_{1} + a_{2}x_{2} + ... + a_{n}x_{n} = b [/tex]

where b is a constant.

My question is essentially asking about the converse of the representation done above: Can we go from the component-wise representation to the point-normal representation?

In other words, can we show that the only vectors normal to the space determined by [tex] a_{1}x_{1} + a_{2}x_{2} + ... + a_{n}x_{n} = b [/tex] are vectors of the form

[itex]t<a_{1},a_{2}...a_{n}>[/itex] where t is a scalar parameter?

BiP

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Unique set of vectors normal to a hyperplane

**Physics Forums | Science Articles, Homework Help, Discussion**