I would want to ask my question by first taking an example.
Consider a vector $\vec{v} = \begin{bmatrix} v_{x} \ \ v_{y} \end{bmatrix}^{T}$ which denotes velocity of an object in two dimension, the standard $XY$ cartesian plane, and lets consider this to be our inertial frame. Now, the components of this vector are in the same inertial frame itself that is to say, the components $v_{x}$ along the x-axis and $v_{y}$ along the y-axis of the inertial frame of reference which is the $XY$ plane under consideration.
Thus, the vector $\vec{v}$ exists in the two dimensional $XY$ plane.
Now, my question is as follows. Condsider the $12 \times 12$ state vector of a quadcopter: $$\vec{X} = \begin{bmatrix}x \ \ y \ \ z \ \ \phi \ \ \theta \ \ \psi \ \ u \ \ v \ \ w \ \ p \ \ q \ \ r \end{bmatrix}^{T}$$ where $\begin{bmatrix}x \ \ y \ \ z \ \ \phi \ \ \theta \ \ \psi \end{bmatrix}^{T}$ correspond to the linear and angular position of the quadcopter in the inertial frame and $\begin{bmatrix} u \ \ v \ \ w \ \ p \ \ q \ \ r \end{bmatrix}^{T}$ in the body frame. The research paper from which the model and the state vector is taken is referenced for your convenience: https://www.kth.se/polopoly_fs/1.588039.1550155544!/Thesis%20KTH%20-%20Francesco%20Sabatino.pdf The above vector is found starting from page 12, section 2.3 of the paper.
Here, the vector $\vec{X}$ lies in both the coordinate frames i.e., the inertial frame and the body frame simultaneously, as the first half vector components of the vector lie in the inertial frame(which is fixed) and the second half vector components of the vector lie in the body frame(which is somewhere else in the space and is related to the inertial frame via a rotational matrix). How is this possible? How can a single vector lie in two frames simultaneously as its counter-intuitive to the example considered in the beginning, where the velocity vector lies only in the single inertial frame?
Is it that in state-space analysis, the vectors can be as abstract as possible and are used only to make the math and notation convenient and is different from the vectors used in physics?
Please help me figure this out. Thank you so much for your time.