Reorienting and resizing a part relative to another

I’m trying to write a function that takes a part of a given rotation and reorients it while also resizing it to keep it’s original appearance.

Lets say given parts are always rotated orthogonally. Is there a clean way to find the relative orientation between them, and apply that rotational offset to the size of the part?

I’ve tried some stuff with LookVector/UpVector/RightVector but it’s not very elegant or versatile.

Example

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