An orthonormal basis is an orthogonal basis of unit vectors. If $q_1,\ldots,q_n$ is such a basis, $v=\sum_jq_j\langle q_j,v\rangle$ and $\|v\|^2=\sum_j|\langle q_j,v\rangle|^2$. In matrix form, the basis matrix is unitary, or orthogonal over $\mathbb R$.
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