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(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.

Glorfindel
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jackb
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1 Answers1

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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.

Rory Daulton
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  • Thanks. It is just that I'm not very familiar with this topic. I've seen some solutions for $n=2,3,4$ in that page, but I wondered how they could be generalized for $n>5$ and how to define such matrices. When I first saw that page, I thought to see different definitions for the same rotation matrices, and I didn't know which one was the most suitable for my needs. – jackb Dec 09 '14 at 06:48
  • So how would you suggest to use such definitions in order to define a rotation that involves all the $n>5$ axes together? – jackb Dec 09 '14 at 06:54
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    It is not clear what your problem is that needs a "solution." Is it to find a rotation matrix that will map given vectors $\vec a$ and $\vec b$ to vectors $\alpha$ and $\beta$ respectively? That could be done by finding vectors perpendicular to all four given vectors then rotating around those vectors. Those perpendicular vectors could be found by listing the 4 given vectors before the usual basis vectors and applying the Gram-Schmidt orthogonalization process. – Rory Daulton Dec 09 '14 at 11:35
  • I'll state my problem here and I'll update my question too; I am interested to find a rotation for $\vec a$ to $(|a|,0,\dots)$ and then I'll apply the found rotation for $\vec a$ to $\vec b$ in order to obtain a new rotated $\beta$.

    Hopefully this rotation will preserve the cosine distance, that is the cosine distance between $\vec a$ and $\vec b$ will be the same of $\alpha$ and $\beta$.

     – jackb Dec 09 '14 at 12:50
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    @jackb: You should start another thread and state your new problem there. Make sure you state it clearly. As far as I can tell, my answer satisfies your original question here, as least as I interpret your question. – Rory Daulton Dec 09 '14 at 13:19
  • Ok, I'll post another thread and delete this one. – jackb Dec 09 '14 at 15:18
  • @jackb: Leave a comment here for me for about a day before you delete this thread, so I can find your new one. If this answer has helped you, remember to "accept" it first. – Rory Daulton Dec 09 '14 at 17:36
  • Ok, sure, added on the top of the thread, here it is and thanks again http://math.stackexchange.com/q/1059383/81298 – jackb Dec 09 '14 at 17:49