$$v_1=(x_1,x_2,x_3\cdot\cdot\cdot,x_n)$$is such a vector. By changing its signs and positions of each component $x_i$, we can get different vectors. When n is odd, it's impossible for any of orthogonal to each other. When n is even, what is the maximum number of vectors that are orthogonal to each other.
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 $n$=$4$.
What about $n=4,6,8,10,...$ and how many?