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Please read whacka's answer to this post since my question is related to the answer.

I would like to solve the problem: If $q \in S^3$, write $q = \cos \theta + I \sin \theta$ with $I \in S^2$. Prove that $\rho (q) \in SO(3)$ is the rotation through $2 \theta$ about the axis $OI$.

$\rho$ is an isomorphism between $S^3/\{\pm1\}$ and $SO(3)$, but this is not important here since I just want to understand the problem itself better regarding $S^2$, the group of points on a unit circle.

According to whacka, $I = \frac{\text{Im}(q)}{||\text{Im}(q)||}$. And, I would like to show that $I \in S^2$.

$I$ is a unit vector in $S^3$, and choose $v \in S^3$ s.t $I \perp v$.

Note that $I,v$ is on the unit circle which belongs to the plane spanned by $I,v$.

Then, $I \in S^2$.

Is this the right approach to understand $I$ as a vector in $S^2$?

john
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1 Answers1

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I think you should do it more concretely. $I$ is a purely imaginary quaternion, so it is written $I=bi+cj+dk\in \mathbb{R}^3$. And since $I=\frac{\text{Im}(q)}{|\text{Im}(q)|}$, $I$ has unit norm, $b^2+c^2+d^2=|I|=\frac{|\text{Im}(q)|}{|\text{Im}(q)|}=1$. Thus $I$ is in $$S^2=\{\text{unit vectors in }\mathbb{R}^3\}$$

Keshav
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