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From what I understood, quaternion rotations could essentially be described as a vector coming from an object's location and going to (x,y,z) and W is a value that determines the object's rotation around that vector:

example_1

In this picture, the black arrow is the vector from the object's location (which I suppose would also be the origin in that object's local space?) to (1,1,1), my (x,y,z) point. The teal arrows circling the vector are essentially w, in that they represent some motion around the vector (they aren't a representation of magnitude though, since for some reason which I'm sure will be elaborated on, when w gets really high or low, it slows the rotation).

What I'm confused about is when I change (x,y,z) to (2,2,2) I have only changed the vector in magnitude, not direction. So I would think that the magnitude of the vector wouldn't effect wouldn't effect the rotation if W stays the same, but this shows otherwise:

example_2

I did some fiddling around and I found out that when (x,y,z) is (2,2,2) and w is 2, it gives the same rotation as when (x,y,z) is (1,1,1) and w is 1:

example_3

So I know, w=a and (x,y,z)=(a,a,a) gives the same rotation. I tried something else, comparing w=1 and (x,y,z)=(1,1,0.5) with w=2 and (x,y,z)=(2,2,1) and finally w=3 and (x,y,z)=(3,3,1.5) for good measure. They had the same rotation as well, so I know that as long as the four values maintain the same ratio, they will have the same rotation.

This where I've hit a bit of a standstill.

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    $\begingroup$ They are 4-dimensional vectors. The relationship is mathematical. I don't think you can make much sense of them by looking at them. $\endgroup$ – Martynas Žiemys Apr 17 '19 at 5:56
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    $\begingroup$ I'd imagine any valid answer to this question would just be a summary of the Quaternion page on Wikipedia, and is more suited for the Math SE site since Quaternions aren't specific to Blender. $\endgroup$ – Michael Glen Montague Apr 17 '19 at 7:40
  • $\begingroup$ Agree this is better suited for maths stackexchange. There are a number of "visualizing quaternions" references. Also it appears you are describing axis-angle defined rotation. $\endgroup$ – batFINGER Apr 17 '19 at 8:26
  • $\begingroup$ I suggest to watch the free YouTube "Humane rigging" lessons by Nathan Vegdahl, where quaternions are explained in an animator's point of view. $\endgroup$ – josh sanfelici Apr 17 '19 at 8:49
  • $\begingroup$ I thoroughly recommend 3Blue1Brown & Ben Eater's interactive demonstration and explanation of the relationship between quaternions and 3-rotations here. You get to play with quaternions, see that you need a quaternion and its inverse actually to perform a rotation, etc.etc. $\endgroup$ – Robin Betts Apr 17 '19 at 9:22

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