# Quaternions and Rotation?

• Mar 20th 2011, 02:41 PM
Conn
Quaternions and Rotation?
I got this question in my Group Theory class and I'm not really sure how to go about it because I never really understood quaternions in the first place!

Let v be the unit vector ($\displaystyle$\frac{1}{\sqrt{3}}$,$-\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$$) and let R be the rotation through angle \displaystyle 60^\circ$$ anticlockwise around Ov

Calculate R(1,0,0) by conjugating a quaternion.

P.S. That one in the first part of the vector should be on top, I couldn't get the MATH tags to wrok properly
• Mar 20th 2011, 03:15 PM
EuropeanSon
You are in my class, it appears. Unless our lecturer steals his questions from elsewhere.

Section 23.4 in his notes cover this question pretty much perfectly.
• Mar 20th 2011, 03:42 PM
Conn
Oh well that's handy what section is relevant to question 3, do you know??
• Mar 20th 2011, 03:46 PM
EuropeanSon
I'm not sure if one is. I posted a thread here earlier looking for an answer, but nobody has responded yet.
• Mar 20th 2011, 03:48 PM
dwsmith
Quote:

Originally Posted by EuropeanSon
I'm not sure if one is. I posted a thread here earlier looking for an answer, but nobody has responded yet.

If you don't show the forum you have tried anything, you won't get much help.
• Mar 20th 2011, 04:19 PM
EuropeanSon
The problem with that is that I have no idea where to start. I don't know how to calculate a unitary matrix from a quaternion (in actuality I have no idea how to form ANY matrix from a quaternion) and so without that basic knowledge can do no work. It's fairly elementary. I'm not asking for the answer to the question, just a piece of basic knowledge from which I can do some work.
• Mar 21st 2011, 01:48 AM
TheArtofSymmetry
Quote:

Originally Posted by Conn
I got this question in my Group Theory class and I'm not really sure how to go about it because I never really understood quaternions in the first place!

Let v be the unit vector ($\displaystyle$\frac{1}{\sqrt{3}}$,$-\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$$) and let R be the rotation through angle \displaystyle 60^\circ$$ anticlockwise around Ov

Calculate R(1,0,0) by conjugating a quaternion.

Every rotation of $\displaystyle \mathbb{Re}^3$, given by the axis u and the angle of rotation $\displaystyle \alpha$, is the result of conjugation by unit quaternions $\displaystyle t=cos\frac{\alpha}{2}+\vec{u}sin\frac{\alpha}{2}$ (see here). Now you have $\displaystyle \vec{u}$, $\displaystyle \alpha=-\frac{\pi}{3}$, and $\displaystyle v=(0, 1, 0, 0)$ in the link. Can you proceed from here?