Rotating an $n$-dimensional hyperplane

Let $\mathcal{H}: \mathbf{x}^T\mathbf{w}+b=0$ be a hyperplane in the $n$-dimensional Euclidean space of column vectors. Is there a way of "rotating" the above hyperplane such that it coincides with the $(n-1)$-dimensional hyperplane $x_2x_3...x_n$ in order the dimensions $x_2,...x_n$ to take values in $(-\infty, +\infty)$ and $x_1\in(c,+\infty)$?

Does it happen using special orthogonal groups in $n$-dimensions (SO($n$))? How exactly does it happen? Which is the value of $c$ above? Is it a change of variable $\mathbf{y} = R\mathbf{x}$, where $R\in SO(n)$? I have studied about special orthogonal groups, but I cannot find the answer...

Any hint on that would be great! Thanks in advance!

• Is there a feasible way to make this question be given more attention? I find this problem interesting, I think that many other people do so, but what can I do? Commented Nov 13, 2013 at 12:09
• Yes! Just give it some bounty for it! I don't have enough credits to give part of it for this but if you collect enough points, you can! Commented Nov 23, 2013 at 18:42
• I am also dealing with a similar problem, therefore I also feel the same; but just imagine how vast is the space of possible problems and you would realise its not strange that people might not be on the same freq as you are. Commented Nov 23, 2013 at 19:07