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

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# Unique set of vectors normal to a hyperplane

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