For 3D Euclidean space, does the metric of space \begin{gather} \delta_{ij}=\mathbb{I}= \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \end{gather} show that the unit vectors are orthogonal \begin{gather} \hat{e_i}\cdot\hat{e_j}=\delta_{ij} \end{gather}

or does the fact that the unit vectors are orthogonal dictate what the metric of space must be?

  • 1
    $\begingroup$ Which came first, the chicken or the egg? $\endgroup$ – Willie Wong Apr 19 '18 at 16:03
  • 3
    $\begingroup$ That's two ways of saying the same thing. $\endgroup$ – John Rennie Apr 19 '18 at 16:10

Orthogonality is practical but not required. As long as the determinant of the matrix does not vanish all is well ...


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