Let me first fix some notation to avoid confusion. Assume I have two frames $A$ and $B$, having basis vector $i_{A}, j_{A}, k_{A}$ and $i_{B}, j_{B}, k_{B}$, respectively, and a rotation matrix $R_{A/B}$ that converts a three dimensional vector represented in the $B$ frame, say $v|_{B}$, to the same vector represented in the $A$ frame, that is
$$ v|_{A} = R_{A/B}v|_{B}$$
Also, assume that the last row of $R_{A/B}$, represented by $R_{A/B}(3)$, is
$$R_{A/B}(3) = \begin{bmatrix}r_{1} &r_{2}&r_{3}\end{bmatrix}$$
Now, if $v|_{A}=\begin{bmatrix}v_{1} &v_{2}&v_{3}\end{bmatrix}$, and $v|_{B}=\begin{bmatrix}w_{1} &w_{2}&w_{3}\end{bmatrix}$, then
$$v_{3} = r_{1}w_{1}+r_{2}w_{2}+r_{3}w_{3}$$
Now my question is: If I convert $v|_{B}$ to a two dimensional vector my taking a norm, say $v_{n}|_{B} = \begin{bmatrix}||w_{1},w_{2}||&w_{3}\end{bmatrix}$, where this new two dimensional vector is now in the 2D frame $B'$, where $k_{B'} = k_{B}$ (similarly for A'), how can I get a rotation matrix from $B'$ to $A'$ that gives me the same $v_{3}$ as before.
By brute force calculation, I can set the third row of the new 2x2 rotation matrix, say $\begin{bmatrix}a & b\end{bmatrix}$, to give me the desired result:
$$ v_{3} = r_{1}w_{1}+r_{2}w_{2}+r_{3}w_{3} = a||w_{1},w_{2}|| + bw_{3} $$
This gives me $a = (r_{1}w_{1}+r_{2}w_{2})/||w_{1},w_{2}||$ and $b = r_{3}$. However, you can notice that my new 2x2 rotation matrix is a function of the vector I am trying to rotate. I want to avoid this, but I don't know how.
My brain keeps going towards axis angle and taking advantage of that somehow, but I have been thinking about this for some hours and my head is spinning. Any advice or direction would be greatly appreciated.