Quaternion Rotations: Show R2∘R1 Is a Rotation

[Lightf]

Homework Statement

3.
[STRIKE](a) Show that every quaternion $$z$$ of length 1 can be written in the
form $$z = cos(\alpha/2) + sin(\alpha/2)n$$, for some number α and some vector $$n$$, $$|n| = 1.$$[/STRIKE]

(b) Consider two rotations of the 3d space: the rotation $$R_1$$ through $$\alpha_1$$ around the vector $$n_1$$ and the rotation $$R_2$$ through $$\alpha_2$$ around the vector $$n_2$$. Define a new transformation, the composition $$R_2 \circ R_1$$, in the usual way: it takes a vector v, rotates it using $$R_1$$, and then rotates the result using $$R_2: R_2 \circ R_1(v) = R_2(R_1(v))$$. Show that this composition is also a rotation around some vector through some angle. (Hint: in quaternionic terms, $$R1$$ brings v to $$w_1v\overline{w_1}$$), and $$R_2$$ brings the result to

$$z = cos(\alpha/2) + sin(\alpha/2)n$$

$$w_2w_1v\overline{w_1}\overline{w_2} = (w_2w_1)v(\overline{w_2w_1})$$

Now, check that $$w = w_2w_1$$ is of length 1, and use the previous problem.)

Homework Equations

$$w_2w_1v\overline{w_1}\overline{w_1}\overline{w_2} = (w_2w_1)v(\overline{w_2w_1})$$.

The Attempt at a Solution

I don't understand the question, if someone could explain what I must do that would be really helpful!

 

[Maybe_Memorie]

Vladimir actually did this out for us in class sometime last week.

You could probably approach this using the method of the optional question from last weeks homework.

(off topic, I know we're in the same math class, but who are you??)

 

[Lightf]

I'll check my notes again and try again ;)

(I'm Fergus - You doing pure maths?)

 

[Maybe_Memorie]

I stopped writing my notes just as he started explaining the rotations.. Typical.

I'm stumped with this tbh.

(I'm Paul - I am, and I know you are too because I saw your thread about the computation quiz ;) )