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I want to compute the angle between vectors by means of the formula given here,

$$ \theta = 2\arctan2 \left( \left|\left| \frac{\textbf{y}}{\|\textbf{y}\|} - \frac{\textbf{x}}{\|\textbf{x}\|} \right|\right|, \left|\left| \frac{\textbf{y}}{\|\textbf{y}\|} + \frac{\textbf{x}}{\|\textbf{x}\|}\right|\right| \right) $$

Provided that I have the coordinates of $\textbf{x}$ and $\textbf{y}$ in the basis $B$. Intuitively makes no sense to explicitly perform the change of basis but I cannot find the proper expression.

Concretely, my problem is,

Given a basis $B=\{ \textbf{u},\textbf{v},\textbf{w}\}$, I want to compute the angle between vectors $\textbf{x}$ and $\textbf{y}$ which are given by,

$$\textbf{x} = \alpha\textbf{u}+\beta\textbf{v}+\gamma\textbf{w}$$ $$\textbf{y} = \delta\textbf{u}+\epsilon\textbf{v}+\zeta\textbf{w}$$

without explicitly perform the change of basis first.

KReiser
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myradio
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  • I'm not sure why you think a change of basis is required. The formula for $\theta$ is already basis independent (as you would expect, since $\theta$ does not depend on the basis chosen). All you need to do is to find $||\textbf{x}||$ and $||\textbf{y}||$. – gandalf61 May 14 '19 at 08:26
  • Yes, but the question is, can I obtain $\theta$ without the basis B? – myradio May 14 '19 at 10:14
  • @gandalf61 Maybe this is not the proper formula to use? Maybe there are some conditions the basis has to fulfil (like being ON)? I am a bit confused but how to I compute $|x|$ and $|y|$ otherwise? – myradio May 14 '19 at 11:01
  • You can’t do this unless the basis is orthogonal. – Algebraic Pavel May 16 '19 at 11:29
  • @AlgebraicPavel Exactly, that's what i was finding. But then I thought does it even need to be orthonormal? Or orthogonal is enough? – myradio May 16 '19 at 13:23
  • Well you don't really need to have the basis but in any case you need at least the gram matrix $B^TB$ to compute the norms and/or dot products from coordinates. – Algebraic Pavel May 16 '19 at 13:32
  • I think I understood now, indeed if the basis is ON the coordinates give the right angle. – myradio May 16 '19 at 21:44
  • I wonder whoever downvoted most likely didn't really understand the question in the first place, or would be good to have any feedback otherwise. – myradio May 16 '19 at 21:45

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