how to calculate dimension of quotient space?

Question

What is the dimension of $O(n) \setminus \mathbb{R}^n$ for $n \geq 2$?

How to calculate this? $n - \frac{n(n-1)}{2}$ does not work well.

Answer 1

The orbits of this group action are vectors with constant norm, that is, $n$-spheres of different radii. Thus, the euclidean norm map $\left\|\cdot\right\| :\mathbb{R}^n\rightarrow \mathbb{R}$ provides a homeomorphism between the quotient space and non-negative reals $\mathbb{R}_{\geq 0}$. So, the quotient is one-dimensional.