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If ` vec e_1, vec e_2, vec e_3a n d vec E_1, vec E_2, vec E_3` are two sets of vectors such that ` vec e_idot vec E_j=1,ifi=ja n d vec e_idot vec E_j=0a n difi!=j ,` then prove that `[ vec e_1 vec e_2 vec e_3][ vec E_1 vec E_2 vec E_3]=1.`

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To prove that \([ \vec{e_1}, \vec{e_2}, \vec{e_3} ] [ \vec{E_1}, \vec{E_2}, \vec{E_3} ] = 1\), we start by analyzing the given conditions. ### Step 1: Understand the inner product conditions We are given that: - \(\vec{e_i} \cdot \vec{E_j} = 1\) if \(i = j\) - \(\vec{e_i} \cdot \vec{E_j} = 0\) if \(i \neq j\) This indicates that the vectors \(\vec{e_1}, \vec{e_2}, \vec{e_3}\) and \(\vec{E_1}, \vec{E_2}, \vec{E_3}\) are orthonormal. ...
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