- #1

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## Main Question or Discussion Point

This has been bugging me for a while, and I feel like I’m missing or misunderstanding some crucial piece of information, so please advise me: the scalar product of two vectors (say ##\mathbf v## and ##\mathbf w##) is given using the metric: ##g_{\alpha \beta} \mathrm v^{\alpha} \mathrm w^{\beta}##. So how can we define the components of the metric, ##g_{\alpha \beta}##, as the scalar product of the basis vectors? Wouldn’t calculating the scalar product of the basis vectors require knowledge of the components of the metric to begin with? Or is "calculating" the scalar product of the basis vectors just nonsense- i.e. you don't calculate them, instead, you choose them, and in so doing, simultaneously choose the components of the metric?