Find an equation of perpendicular bisector between two points in higher dimensions

Question

I am working on something that requires me to find a reflection matrix in very hight dimensions $2^{n}$ for $n>3$. I need that reflection matrix to send point $a$ to point $b$, which are known to me. I thought of the idea of perpendicular bisectors used in 2 dimensions, and wanted to know if there is an idea similar to this that can be used in higher dimensions. If so, do you have any suggestion how to find that matrix, or at least an equation for the hyperplane?

Answer 1

Let $v=a-b$ and $p=(a+b)/2$ the midpoint between $a$ and $b$. The equation of the hyperplane orthogonal to $v$ through $p$ is $$ v\cdot(x-p)=0. $$