# Thread: Quaternions and Rotation?

1. ## 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 ( $\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$)

and let R be the rotation through angle $60^\circ$ anticlockwise around Ov

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

Thanks for any help you can give me!

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

2. 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.

3. Oh well that's handy what section is relevant to question 3, do you know??

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

5. 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.

6. 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.

7. 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 ( $\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$)

and let R be the rotation through angle $60^\circ$ anticlockwise around Ov

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

Thanks for any help you can give me!

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

Note also that unit quarternions described by matrix (a+bi,c+di;-c+di,a-bi)=a1+bi+cj+dk, where 1=(1,0;0,1), i=(0, -1;1,0), j=(0, -i;-i,0), k=(i,0;0,-i) forms a group under multiplication which is isomorphic to SU(2) (see here).