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"let $ a,b \in \mathbb{R}$ and $t \subseteq \mathbb{R}$

$a \equiv b (mod. 2\pi)$ if $\exists k \in \mathbb{Z}(a-b=2k\pi)$

this congruence is equivalence relation, therefore:

$[a]_\sim=\{x|x=a+2k\pi \wedge k \in \mathbb{Z}\}$

$t$ is angle if $t \in \mathbb{R} / \sim$

but, let $\vec{v}\neq\vec{0},\vec{w}\neq \vec{0}$ and $\vec{v},\vec{w} \in \mathbb{R}^2$ (and $\mathbb{R}^2$ is Euclidean plane), How do I define the angle between $\vec{v}$ and $\vec{w}$?

Thanks in advance!!

mle
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  • You cannot define the angle of two vectors as a value in $\Bbb R/2\pi\Bbb Z$ unless one is in dimension$~2$ (and the vectors are nonzero). Your question is not clear about whether you are only talking about vectors in the Euclidean plane or not; please be specific. – Marc van Leeuwen Nov 23 '13 at 12:53
  • @MarcvanLeeuwen, I correct the post...!! Thanks! :) – mle Nov 23 '13 at 12:59
  • See also the related question http://math.stackexchange.com/q/583066/9754 – davidlowryduda Dec 04 '16 at 19:07

1 Answers1

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By definition the (oriented) angle from $\vec v$ to $\vec w$ is the unique value $\alpha\in\Bbb R/2\pi\Bbb Z$ such that $R_\alpha\cdot \vec v$ is a positive scalar multiple of $\vec v$ (or equivalently $R_\alpha\cdot \frac{\vec v}{|v|}=\frac{\vec w}{|w|}$) where $R_\alpha$ is the rotation $$ R_\alpha=\begin{pmatrix}\cos\alpha&-\sin\alpha\\\sin\alpha&\cos\alpha\end{pmatrix}. $$

Marc van Leeuwen
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