Orientation Differences between Coordinate Frames

Hi-
  I’m looking for an easy way to determine rotational differences between coordinate frames in 3D space.

My data looks like this:

Coord1=[X Y Z RotX RotY RotZ]
X, Y, Z => location of Coord1 using a Global coordinate frame for reference
RotX, RotY, RotZ=> orientation of Coord1, defined as a series of rotations from the global coordinate frame

If I want to look at translational differences between Coord1 and my Global frame, I would use the distance formula:

3Ddistance=Sqrt(X^2+ Y^2+ Z^2)

BUT I want to look at rotational differences!
RotX, RotY, and RotZ describes the rotational difference with respect to each Global axis. So, what would be the rotational equivalent of the distance formula? I want one number describing the overall rotational error between Coord1 and my Global frame.
I imagine it as taking the vector equidistant from each Coord1 axis and comparing it to the vector equidistant from each of the Global’s axes. What would be the rotational difference between them?

I’m having a hard time visualizing these calculations and would love suggestions.
Are there any toolboxes or functions that could assist me in this?

Thanks in advance for your help!