I'm having confusion in regarding how to compute the new coordinates of a point in 3D, when a 3D rotation occurs along the Z-axis. Consider all the systems are right-handed.
In the book, "Quaternions and Rotation Sequences by Jack B.Kuipers", the following approach is given :
(1) Rotation Matrix Used in Book
But I have noticed that in MATLAB and many online 3D Rotation Calculators, the transposed version of the above matrix (as in (1)) is used and this yields correct results. The matrix that gives correct results is :
(2) Correct Rotation Matrix to be used
So, I don't understand the mismatch in notations and I'm confused about which among the above-mentioned matrices is the correct approach ?
One more doubt I had is which is the rotation matrix among (1)/(2) is to be used in Method-1 and which is to be used in Method-2 ?
Method 1 :
row [new coordinates] = row [old coordinates] * [Rotation Matrix]
Method 2 :
column [new corodinates] = [Rotation Matrix] * column [old coordinates]
I'm guessing that the rotation matrices in methods 1 and 2 are transpose of each other. But I'm not sure as to which rotation matrix among (1)/(2) is used in method 1 and which in method 2.
Use Matrix 1 for method 1.
Use Matrix 2 for method 2.
Note that method 1 $x' = Rx$ when transposed becomes
$(x')^T = (\mathbf Rx)^T = x^T \mathbf R^T$ (Method 2).
It is really helpful to try examples for yourself that you can compute by hand and convince yourself what the matrices are doing.
It is also really important to understand the three interpretations of a rotation matrix:
It represents the orientation of one coordinate system with respect to another.
It represents a coordinate transformation (passive picture) in which a fixed point is converted from one coordinate system to another. I.e. the point stays fixed and the coordinate systems change. We change the representation of the same point.
It represents a point transform (active picture) in which a point moves in a fixed coordinate system. i.e. the axes stay fixed and the point changes.