Guys, the unit sphere in $\mathbb R^n$ is defined as
$$
S^{n-1}=\{x\in\mathbb R^n:\Vert x\Vert=1\}.
$$
Why is it $S^{n-1}$ and not $S^n$? I'm guessing it has something to do with dimensions. When we look at the unit circle, $S^1$, it's clear that $S^1$ contains lines (which have dimension 1) - however, I can't quite see it in the generally case.