Rotating a $\vec{v}$ by a unit quaternion $\vec{q}=\vec{u}$, we have the same vector. So, rotation does not change it.
$$R(\vec{v}) = \vec{q}\vec{v}\vec{q}^*$$ $$=(cos θ + \vec{u} sin θ)\vec{v}(cos θ − \vec{u}sin θ)$$ $$=((cos θ)^2 - \vec{u}^2 (sin θ)^2)\vec{v}$$ $$=((cos θ)^2 + (sin θ)^2)\vec{v}$$ $$=\vec{v}$$
So, i am fine with this result.
But, for example when we rotate a (0,2,0,0) with a unit quaternion (0.707,0,0,0.707) (axis:(0,0,1) and angle $\pi/2$), we get result vector(0,0,2,0) which is different(!) from the original vector.
So i am confused how these vectors are equal! Algebra is fine but arithmetic(if ever could be called like this) confused me. I can accept algebra but I cannot give up proof without induction from examples.
Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal. (But same magnitude and direction should not mean coordinate equality because of the sample rotation.)
So, the norm of vector does not change after unit(norm 1) quaternion rotation but i am not sure about directions.
We could compute direction cosines but how to prove overall direction is not changed?
Also, i am a little bit confused about what does a $\vec{v}$ represent intuitionally; so there should be vectors with different x,y,z components but represent same vector?