Need to understand: derivating euler rotation from fixed axis rotation

[muffinman123]

Homework Statement

Homework Equations

given in question

The Attempt at a Solution

I already know the solution but I don't understand what it means

the process is simply fixed(c,b,a) to euler(a,b,c)

Rz(a)=Rz(a)
Ry(b)=Rz(a)Ry(b)Rz'(a)
Rz(c)=Rz(a)Ry(b)Rz(c)Ry'(b)Rz'(a)

write it in fixed form (c,b,a)

Rz(c)Ry(b)Rz(a) = blah blah, and you get Rz^(a)Ry^(b)Rz^(c)

BUT forget all that

I want to know how the similiarity transoform looks at certain of middle step of a rotation as though it's being observed in fixed axis, and why is the question answered the way it is. I thought I was supposed to multiply euler rotation Rz(a)Ry(b)Rz(c) and try to reach fixed axis rotation Rz(c)Ry(b)Rz(a), but the question clearly asks me to do it the reverse of that because the similiarity transform only let's me get it that way.

The book does not explain how that Similiarity Transformation works reverse the rotation or how it's derived. it just brush it off as something obvious while it just barely talked about rotational matrix and reference frames by the end of chapter 2.

This question is fine if you just write the formulas down and forget about it, but I feel I am skipping a lot more without knowing the meaning behind the formulas.

this question is from John Craig's robotics book which has useful stuff in the 1st few chapters but for some of the questions at the end of the chapter, he didn't give me enough explanation to do them and left me puzzled.

 

[LightHero]

I'm not asking for the answer - I already know that. I just don't understand why it works or how the similarity transformation lets me reverse the rotations.