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How do the algebraic and geometric formulas of dot and cross product relate to each other? I have seen the proof for 2 dimensions but how do we generalize it to $n$ dimensions? Is there a way to prove it without the cosine law? That proof feels like a chicken and egg proof as some use vectors to prove cosine rule and vice versa.

Proof for 2 dimensions: Proof of equivalence of algebraic and geometric dot product?

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Runjia Chen
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  • $n=2$ is enough; the identity holds (appropriately formulated) for every $2$-dimensional subspace of $\mathbb{R}^n$, so just consider the subspace spanned by the two vectors in question. – While I Am Jan 09 '22 at 05:17
  • I don’t think you can generalize to $n$ dimensions. For one thing, the cross product doesn’t make sense in $n$ dimensions (for most values of $n$). Generalizing to $n=3$ dimensions makes sense. But, as the other comment points out, you’ll really be working in the 2D sub space spanned by the two vectors. – bubba Jan 09 '22 at 06:21
  • Sorry, how do you generalise the proof of equivalence for dot product to n dimensions? – Runjia Chen Jan 09 '22 at 06:27

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