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 ;) )