From a part on proving the Hessian criterion for the existence of minima and maxima in the book Vector Calculus (Colley S.).
It states that in $R^n$, the set of all unit vectors forms an $N-1$ dimensional sphere $S$, which is compact.
Please elaborate why the set of all unit vectors form an $N-1$ dimensional sphere rather than $N$ which seems more appropriate to my imagination.