# Is every hyperplane in $\mathbb{R}^n$ determined by a unique normal vector?

Is every hyperplane in $\mathbb{R}^n$ determined by a unique normal vector? And why?

I analysed for $\mathbb{R}$, a hiperplane in $\mathbb{R}$ is a point, so the hyperplane is $PX= \alpha$, with $\alpha \in \mathbb{R}$. So $X=\alpha / P$.

• The normal vector isn't unique. But the span of any normal vector will be the same one-dimensional subspace. To prove this you use that theorem that says that you can always complete a basis of a subspace to a basis of the ambient space. Then you just prove that any such extension for a basis of your hyperplane will be made by adding a vector determined up to scalar multiplication (you can do this part by proof by contradiction).
– user137731
Nov 20 '15 at 0:30
• If you choose a unit normal vector it's almost unique-choosing the sign requires some additional information, namely, an orientation. Nov 20 '15 at 0:44

No. Consider $\mathbb{R}^3$, and the (hyper)planes $x = 0$ and $x = 1$. Both have $[c, 0, 0]$ as normal vectors for all $c \neq 0$. So they are neither determined by their normal vectors, nor is their normal vector unique.