Suppose we have a set of vectors $\{e_1, e_2, \ldots, e_m\}$ in a Euclidean space $\mathbb E^n$ such that for all distinct $i$ and $j$, the inner product satisfies the following property: $\langle e_i, e_j\rangle\in[-1; 1]$.
Does there always exist a set of unit vectors $\{e'_1, e'_2, \ldots, e'_m\}$ in $\mathbb E^n$ such that for all distinct $i$ and $j$, the inner product satisfies the following property: $\langle e_i, e_j\rangle = \langle e'_i, e'_j\rangle$?
No, not always. Consider in $\Bbb{R}^n$ the three vectors: \begin{align*} e_1 &= (1, 0, 0) \\ e_2 &= (-1, 0, 1) \\ e_3 &= (-1, 0, -1). \end{align*} Then \begin{align*} e_1 \cdot e_2 &= -1 \\ e_1 \cdot e_3 &= -1 \\ e_2 \cdot e_3 &= 0. \end{align*} If there were unit vectors $e'_1, e'_2, e'_3$ with the desired properties, then $$e'_1 \cdot e'_2 = e'_1 \cdot e'_3 = -1,$$ which would imply that $e'_2 = e'_3 = -e'_1$. We would therefore be forced to have $e'_2 \cdot e'_3 = 1 \neq 0$.
