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Largest set of perpendicular vectors I am curious of this original questions when I first saw it and I thought it's kind of combinatorics. I am learning both linear algebra and combinatorics. This question is kind of interesting.

Find the maximum numbers of component of vectors of a given $n$ dimensions such that they are perpendicular to each other.

The condition is that the component in each vector has to be a permutation of of the original one

The original vectors:$e_1=(x_1, x_2......x_n)$

all the permutations:$e_2=(x_9, -x_2......x_n)$, $e_3=(-x_4, x_{12}......-x_n)$......

For example, vectors $(x_1, x_2, x_3, x_4)$, $(-x_2,x_1,-x_4,x_3)$ and $(x_3,x_4,-x_1,-x_2)$ are mutually perpendicular when $n$=$4$

When $n$ is odd, all the vectors formed by the permutation of the component of the vector are impossible to be mutually perpendicular, but I don't know the proof.

When $n$ is even, there are definitely many vectors which formed by the permutation of the components of the original vectors are mutually perpendicular.What's the pattern of the perpendicular?

I tried myself and find that the vectors of different combinations of $+$ and $-$ signs are always mutually perpendicular when $n$ is even.But I can't prove it.

With enumeration, it's easy to find the pattern, but where is the proof.

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    This is slightly nit-picky, but I wouldn't call $(-x_2, x_1, -x_4, x_3)$ a permutation of $(x_1, x_2, x_3, x_4)$; it's a bit more complicated than that. You're letting the symmetry group of the 4-dimensional cube act on vectors, not just the symmetric group. – 2017-01-24
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    can you explain this more detailedly – 2017-01-25
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    what is symmetry group – 2017-01-25
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    Also posted to MO, http://mathoverflow.net/questions/264276/maximum-of-orthogonal-vectors – come on, Mclalalala, you know better than to do this. – 2017-03-10

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