It seems to be idiot, I Wonder "The one rotation can make two random-different 3d coordinate system match." I searched and learned about Quaternion, and i realized one quaternion (length is 1) can contain one rotation info. but i can't understand their multiplication also can be interpreted as ONE ROTATION. If above is right, All multiplication between quaternion of rotation means a simple rotation. but not, All I have learned becomes complex to me.
I know it is right and possible intuitively, I want proof (as easy as possible)
A coodrdinate system of a normed vector space is established by a choice of ordered basis. By "rotation [...] matches one to another" I take it to mean that the rotation would have to map one ordered basis onto the other ordered basis.
With this understanding, the explanation goes this way.
Firstly, "rotation" means that angles, lengths and orientation will be preserved after the transformation. This means you can't map a basis element onto one of a different length, the angles between the basis vectors have to match before and after, and the orientation of the two bases have to be the same.
But if the two chosen bases match in all these respects, then yes, there is one, and only one, rotation to transform one to the other.