Locality-sensitive hashing random projection
No.  The statement you're reading is correct.  Try working through an example (in 2 dimensions, i.e., $d=2$); pick specific values of $v$ and $r$, draw them on the picture, and see what happens.  The set of points $v$ such that $v \cdot r = 0$ is a line; the set of points $v$ such that $v \cdot r \ge 0$ is a half-plane (e.g., the half-plane above that line).