Alignment of Orthonormal Families by Plane Rotations

Let $n$ and $p$ be natural numbers and let $w_1,\dots,w_p$ be an orthonormal family in the Euclidean space $\mathbb{R}^{n}$, with standard basis vectors $e_1,\dots,e_n$. Then:

1. $p\le n$.

2. There exist a finite composition of plane rotations $h$ of $\mathbb{R}^{n}$ and a sign $\varepsilon\in\{1,-1\}$ such that

3. If $p<n$, then $h$ can be chosen with $\varepsilon=1$, so that $h(w_k)=e_k$ for every $1\le k\le p$.

The sign in part 2 cannot in general be removed when $p=n$: for $n=p=2$ the orthonormal family $(e_1,\,-e_2)$ is not mapped to $(e_1,e_2)$ by any finite composition of plane rotations.