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Is it possible to form a tensor (i.e. matrix) such that this tensor applied to the apparent wind vector produces a vector representing the resulting aerodynamic force?

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Yes, actually the vector of the nondimensional aerodynamic coefficients is related to the nondimensional forces using a coordinate transformation tensor. Then, this tensor depends on your coordinate system (stability axes, wind axes) which in turn depends on which coordinate system is more convenient for your job. For example \begin{gather} \begin{Bmatrix} C_{D_i} \\ C_Y \\ C_L \end{Bmatrix} = \Bigg[ \bar{\bar T^s} \Bigg] \begin{Bmatrix} \overline{F}_x \\ \overline{F}_y \\ \overline{F}_z \end{Bmatrix} \end{gather} where $\bar{\bar{T^s}}$ is your stability axes coordinate transformation tensor which incorporates the sines and cosines of the rotation angles and the right hand side vector is the nondimensional force components. For detailed derivations regarding this coordinate systems and their use you may want to check Chapter 6 of the book of Mark Drela, $\textit{Flight Vehicle Aerodynamics}$ from MIT Press.

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