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

Question

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 $X=\alpha / P$.

Answer 1

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.

Note that these vectors comprise all their normal vectors, so there does not exist a normal vector which can distinguish these two planes.