(Moved to https://math.stackexchange.com/questions/1059383/searching-for-a-definition-for-n-dimensional-rotation-which-is-cosine-distance-i, flagged it to delete it)
I'm wondering if there exists a rotation definition by which the vectors $\vec a$ and $\vec b$ keep the same cosine distance even after an $n$-dimensional rotation that transforms $\vec a$ to $\alpha$ and $\vec b$ to $\beta$.
I've tried to search in here for some questions for n-Dimensional rotations (like here, here or here) but in my opinion the most detailed definition that I found was this one:
Given $n\in\mathbb N$ and two vectors $\vec v$ and $\vec w$ where $\vec v,\vec w\in\mathbb R^n$, in order to obtain $M\vec v=\vec w$ where $M$ is defined as $RS$:
$$R(x)=(I - 2uu')x$$ $$S = I - tt'$$
where $\langle t, \vec v\rangle=0$, $\langle t,\vec w\rangle=2$ and $\|t\|=1$, while $u=\frac{\vec v-\vec w}{\|\vec v-\vec w\|}$.
I am wondering if, given two vectors $\vec a$ and $\vec b$ which have cosine distance $k$ (i.e. $\frac{\vec a\cdot\vec b}{\|\vec a\|\|\vec b\|}=k$), and given $RS$ the matrix rotation by which $\vec a$ is rotated to the vector $\alpha=(\|a\|,0,\dots,0)$, I want to apply the same rotation to $\vec b$ in order to have as a result $\beta=RS\vec b$. Does this kind of rotation preserve the cosine distance, and hence does $\frac{\alpha\cdot\beta}{\|\alpha\|\|\beta\|}=k$ hold?
If for this transformation the condition doesn't hold, I'm wondering if it is possible to define such kind of n-dimensional rotation. Thanks in advance.
What is wrong with the common definition that a rotation in $\mathbb R^n$ is a linear transformation whose matrix is orthonormal and has a determinant of $1$?
It can be proven that such a matrix preserves distances, and that combined with the law of cosines can show that the "cosine distance" is also preserved.