I have the following problem: given a vector $vinmathbb{R}^n$ (if it matters, its values are only from $pm1$) and another vector $uinmathbb{R}^n$, I want to project $u$ into the upper-half of the orthogonal complement of $v$.

I have tried to first project $u$ into the entire orthogonal complement of $v$, but now im stuck, since I cant guarantee it will be in the upper-half of it. I have tried various things (for example, projecting $u$ into a the orthogonal complement of $v’$ where $v’=v$ in all but one position, which in it $v’_i=0$, and after the projection I move $u$ in the $i$‘th dimension, so it will be in the upper side of this particular dimension), but none of the methods I have tried managed to do the trick.

I would be glad to get some help!