The real projective space $\mathbb RP^n$ of dimension $n$ is the quotient of $\mathbb R^{n+1}$ by the equivalence relation $\sim$ given by $x\sim y$ if and only if $x=-y$. This amounts to considering $\mathbb RP^n$ as the set of lines through the origin in $\mathbb R^{n+1}$, or the unit sphere $S^n$ where we identify antipodal points $u$ and $-u$.