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According to this article on angles, we can define rotation without using angles, and then use rotation to define angles. The relevant paragraph is at the very end:

But what is a rotation? Is it possible to define a rotation without first introducing the angle of rotation? Yes, this is possible based on the notion of geometric transformation. Rotation is a geometric transformation with a fixed point that preserves distances..

Question: May I have a reference (textbook or paper) regarding how this can be done? My background in this topic is just elementary geometry and linear algebra.

Jean Valjean
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  • One slick approach is via complex algebra. Specifically, one can represent the points on the sphere via complex numbers and then equate rotations to a particular class of transformations. One would still use angles to specify a particular transformation, but this isn't necessary for the definition. (If you're curious, take a look at my answer to this recent question for the flavor of it.) – Semiclassical Aug 20 '14 at 19:45

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Any non-zero angle less than $180^\circ$ can be the angle between two vectors, defined on a plane spanned by those vectors. The matrix of the linear transformation that represents rotation around such angle is the subject of this question. Then, this answer provides a neat way of finding such a matrix in $\mathbb{R}^3$.

Community
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Kuba hasn't forgotten Monica
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  • I was hoping to define rotations without using the idea of angles. If I understood correctly, your answer requires sine and cosine of an angle? – Jean Valjean Aug 21 '14 at 17:44
  • @JeanValjean Yes, but those are obtained only by manipulating the vectors, not angles, and no trigonometric functions themselves are recomputed. The components of the vectors are the directional cosines, after all. – Kuba hasn't forgotten Monica Aug 21 '14 at 18:38