Finding roll/pitch/yaw for rotating local vector to another vector

Although I looked into similar questions regarding this subject, I could not make the link to my situation so correct me if the answer is already on StackExchange…

My starting point is a global xyz-frame and a local FLU-frame (airplane) which is aligned with the xyz-frame (F/x, L/y, U/z). The local FLU-frame does roll/pitch/yaw using the matrix defined at Wikipedia, Tait–Bryan angles $Z_1X_2Y_3$. What I need is the rotation matrix that rotates a local normalized vector $V$ in the FLU-frame to a global normalized vector $W$ in the xyz-frame. The application is that the local vector $V$ can be defined on the airplane and then the whole airplane is rotated such that $V$ ‘looks at’ coordinates $W$. So we have $W=RV$, with $V$ and $W$ known and $R$ the rotation matrix to be determined. Oh, next step is that I want to rotate the rotated airplane around that vector $W$ afterwards, but first get this step working.

Some things I tried or thought of:

• Skipping one rotation to make life simpler is not possible, in many cases you need all three.
• Just calculating a matrix which transforms $V$ to $W$ (divide) does not work because it does not result in a rotation matrix.
• I tried to find the roll/pitch angles such that the transformed U-axis has the same angle with $W$ as with the transformed $V$ so an additional yaw-rotation (around the transformed U…) then suffices, but it becomes really messy quite soon with unclear conditions for the required roll and pitch solutions. Or my analysis approach was wrong…
• stuck…