Intuitively, I know that if $e_1$ is a standard basis vector in $\mathbb{R}^n$, than multiplying $e_1$ by elements of $SO(n)$ rotates $e_1$ around the $(n-1)$-dimensional sphere.
How do I prove, or at least mathematically motivate, this?
2026-03-25 06:34:59.1774420499
Dimension of orbit of a one-dimensional vector under action of $SO(n)$
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