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## Homework Statement

3.

[STRIKE](a) Show that every quaternion [tex]z[/tex] of length 1 can be written in the

form [tex]z = cos(\alpha/2) + sin(\alpha/2)n[/tex], for some number α and some vector [tex]n[/tex], [tex]|n| = 1.[/tex][/STRIKE]

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

[tex]z = cos(\alpha/2) + sin(\alpha/2)n[/tex]

[tex]w_2w_1v\overline{w_1}\overline{w_2} = (w_2w_1)v(\overline{w_2w_1})[/tex]

Now, check that [tex]w = w_2w_1[/tex] is of length 1, and use the previous problem.)

## Homework Equations

[tex]w_2w_1v\overline{w_1}\overline{w_1}\overline{w_2} = (w_2w_1)v(\overline{w_2w_1})[/tex].

## The Attempt at a Solution

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

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