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I have hyperplanes in $\Bbb R^n$ and the angles between any 2 $i,j$ of them with $\alpha_{ij}$. I want to know all their normal vectors. How to do it systematically?

My try: Let $$ R=(\begin{matrix} cos(\alpha_{ij}) & -sin(\alpha_{ij}) \\ sin(\alpha_{ij}) & cos(\alpha_{ij})\\ \end{matrix}) $$ Let all normalvectors be $(1,0,...,0)$. Fix $n_1$. For the $n_2$ one has the angle to the first one $\alpha_{1,2}$ so apply $R_{ij}(\alpha_{1,2})$ to the components $1,2$ of $n_1$. Fix it. For the third one apply R($\alpha_{1,3}$) to components (1,2), but now what? I do not want to change $\alpha_{1,3}$ so I can rotate around $n_1$ as to modify the angle $\alpha_{2,3}$ but how to make it the right angle and can I even get every angle like this?

Also this approach only works when I have always 1 more dimension to rotate in.

Edit: just needed to solve the system of equations $\vec n_i \vec n_j = cos(\alpha_{ij})$ for all $i,j$ combinations while letting $n_1=(1,0,0,..)$

myelf
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  • How are you representing the angle between them? The angle between two planes is not enough to determine a parallel family of planes (even in three dimensions). – Michael Burr Jun 10 '15 at 16:47
  • I am not sure I understand. The construction is unique is not so important to me. Say in 3d all planes go through the origin and the first one is the (y,z) plane, now there are many planes that have angle $\alpha$ with the first plane. I can pick one of them and change it later if needed (by rotating around the x axis). – myelf Jun 10 '15 at 17:31
  • If that didn't help maybe look at http://math.stackexchange.com/questions/735679/what-is-the-coxeter-diagram-for in the top voted answer where it shows the picture of the 3 planes and the paragraph says "I don't really know how to go about finding planes that have the specified relationship..." thats what I am trying to do. – myelf Jun 10 '15 at 17:31

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