FFT of resultant of signals If one has two signals (say, two acceleromters mounted perpendicularly) and a piece-wise resultant acceleration signal is determined, it appears that frequency content information cannot be determined on the resultant signal using an FFT. 


Can someone comment on why this is so? 


It may be that the process of determining the resultant converts the negative content in the individual signals through the process 
$$r(i) = \sqrt{x(i)^2+y(i)^2}$$ which "loses" information (not sure if that is the best way to describe it). This same effect can be seen if one takes a single acceleration signal and processes the FFT of the absolute value of that signal. 


Is there any way of still being able to assess frequency content of a resultant vector? 
Is it even worth pursuing?
First, you should note that the FFT refers to an algorithm for the fast computation of the DFT taking advantage of its symmetry. 

The signals you have are:



where $\mathbf{a}_y$ and $\mathbf{a}_x$ can both have negative a positive values depending on the direction along the $x$ and $y$-axis. These signals are the output of your accelerometers.

What you call "a piece-wise resultant acceleration", calculated like 
$$r(i) = \sqrt{x(i)^2+y(i)^2},$$ is just $\rVert \mathbf a\lVert$, the magnitude of a vector quantity $\mathbf{a}$, which doesn't show the direction.

From NASA webpage, about vector components:


  Why go to all this trouble? Because, in aerospace, we are often
  dealing with forces and forces are vectors. Breaking a single vector
  force into several components allows us to study the resulting motion
  much more easily.


In conclusion, performing a DFT on your vector quantities $\mathbf{a}_x$ and $\mathbf{a}_y$ will give you frequency information both for about magnitude and the direction together, while performing a DFT on your scalar vector $\rVert \mathbf a\lVert$ will just give you the frequency domain of the magnitude, so yes, you are loosing information here.