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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=j` and `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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AI Generated Solution

To prove that \([ ( \vec{e_1}, \vec{e_2}, \vec{e_3} )][ ( \vec{E_1}, \vec{E_2}, \vec{E_3} )] = 1\), we will use the properties of the dot product and the definition of the scalar triple product. ### Step 1: Understand the given conditions We are given two sets of vectors \(\vec{e_1}, \vec{e_2}, \vec{e_3}\) and \(\vec{E_1}, \vec{E_2}, \vec{E_3}\) such 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 means that the vectors \(\vec{e_i}\) and \(\vec{E_j}\) are orthonormal. ...
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