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It is easy to prove in 2D that $v_1\cdot v_2=|v_1||v_2|\cos(\theta)$ where $\theta$ is the angle between $v_1$ and $v_2$.

But how to generalize? What is the proof in n-dimensions?

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    $\begingroup$ The two vectors span a two dimensional subspace. $\endgroup$ – Charlie Frohman Apr 23 at 14:09
  • $\begingroup$ @CharlieFrohman ...usually. $\endgroup$ – Theo Bendit Apr 23 at 14:12
  • $\begingroup$ What definition of the dot product are you starting from? The sum of the products of the terms? $\endgroup$ – Deepak Apr 23 at 14:17
  • $\begingroup$ Here is how the dot product: $ \vec{v} \cdot \vec{w}= v_1*w_1 + v_2*w_2 + ..... + v_n*w_n$ $\endgroup$ – CSPassionant Apr 23 at 14:24
  • $\begingroup$ How do you define the angle between two vectors in $n$ dimensions? $\endgroup$ – md2perpe Apr 23 at 14:33
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This proof uses the Law of cosines to prove it for any number of dimensions.

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