Projection in Euclidean space on hyperplane

Question

Two vectors in the Euclidean space lie on the one side of the given hyperplane. The angle between these vectors is obtuse. Is it true that the angle between their orthogonal projections on this hyperplane is also obtuse?

Answer 1

Let those vectors be $v=u+w$ and $v’=u’+w’$, where $u,u’$ are orthogonal to the hyperplane and $w,w’$ are the projections of $v,v’$ in the hyperplane. Calculate the dot product of $v$ and $v’$:

$$v\cdot v’=(u+w)\cdot (u’+w’)=u\cdot u’+w\cdot w’$$

Now: if $u\cdot u’\gt 0$ ($v$ and $v’$ are on the same side of the hyperplane), but $v\cdot v’\lt 0$ (the angle between $v$ and $v$ is obtuse), then we must have $w\cdot w’\lt 0$, i.e. the angle between $w$ and $w’$ must be obtuse too.