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As you all know, there is geometric place of points of cross of two planes (given as plane vectors) explicitly written simply as $\mathbb{r}=c_1\mathbb{n_1}+c_2\mathbb{n_2}+\lambda\mathbb{n_1}\times\mathbb{n_2}$

More general is $x^i = c_1 n_1^i + c_2 n_2^i + \lambda \epsilon_{ijk} n_1^j n_2^k$, where i,j,k denotes dimension, {x,y,z} for $R^3$, $\epsilon$ is standard form of cross product of vectors, which is actually antisymmetric operator, rotating variables.

But, in more then 3 dimensional space, there is no rotational operator acting $R^3 \rightarrow R^3$. Instead, absolutely antisymmetric operator acts like cyclic index rotation group $Z_n$. This is obvious from excluding every $i$-th component, like first here:

$b_1 a_2 x_2 + b_1 a_3 x_3 + ... = 0 \\ a_1 b_2 x_2 + a_1 b_3 x_3 + ... = 0$

So I was thinking, what is such as elegant expression for n-dimensional intercrossing of two planes?

sanaris
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  • How do you represent a plane in $n$ dimensions? Or are you talking about a hyperplane, represented using its normal vector $n$? But in that case, the result wouldn't be a line but instead some higher-dimensional flat. – MvG Dec 16 '14 at 08:24
  • For intersecting $n-1$ hyperplanes in $\mathbb R^n$ and describing the resulting line there is a suitable generalization of your formula with its cross product, but it involves $n-1$ operands, not two. – MvG Dec 16 '14 at 08:30

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