Comparing "The" vs. "A" Rotation Matrix in Shankar 12.4.4

[bjnartowt]

Shankar 12.4.4 - "the" rotation matrix vs. "a" rotation matrix (tensor operators QM)

Homework Statement

My question comes up in the context of Shankar 12.4.4. See attached .pdf.

Homework Equations

See attached .pdf

The Attempt at a Solution

See attached .pdf

I have this problem: on Shankar p. 313, they say:
>>>>
We call V a vector operator if V's components transform as components of a vector under a passive transformation generated by U[R]",

${U^\dag }[R]{V_i}U[R] = {R_{ij}}{V_j}$

where R[ij] is *the* 2x2 rotation matrix appearing in [12.2.1] (NOTE, below)...The same definition of a vector operator holds in 3D as well, with the obvious difference that R[ij] is a 3x3 matrix."
<<<<<<<<<<<<<<<<
But aren't there multiple 3x3 rotation matrices?

Also: I have attached some work from Merzbacher, beginning of chapter 12. I did this work, and something isn't clicking in my thick skull...



NOTE: Can't look up 12.2.1, because p. 306 of my .pdf book is gone, and I didn't bring my hard-copy book home, as I'm on Thanksgiving break).

 

[bjnartowt]

I'm sorry, I didn't make my question clear. Initially, I am asking: *what* R[ij] is being talked about as "The" R[ij]. I think this is somewhat-related to my "another another" try in the first attachment "320 - pr 4-4 - classical-esque...", where I try to reverse-engineer a matrix R that has the components I want. However, as my aim is to solve this problem correctly, that may not even be the right question to ask...

 

[bjnartowt]

...It also looks like the .pdfs didn't go through -_-