Let $n\in\mathbb{N}$ with $n\geq 1$, let $I_n\in M_n(\mathbb{R})$ denote the identity matrix, and let $O(n):=\{B\in M_n(\mathbb{R}) : B^\top B=I_n\}$ be the [orthogonal group](/page/Orthogonal%20Group). Let $A\in M_n(\mathbb{R})$, and let $v_i\in\mathbb{R}^n$ denote the $i$-th column of $A$ for each $i\in\{1,\dots,n\}$. Then $A\in O(n)$ if and only if $|v_i|=1$ for every $i\in\{1,\dots,n\}$ and $\langle v_i,v_j\rangle=0$ for all distinct $i,j\in\{1,\dots,n\}$.