A Rotation matrix and rotation of coordinate system

AI Thread Summary
Changing the orientation of a coordinate system involves using a rotation matrix, represented as x' = Ax, where A is defined by specific trigonometric functions of the Euler angles. The matrix A is orthogonal with a determinant of +1, confirming it functions as a rotation matrix that rotates vectors. When A indicates a clockwise rotation by an angle α, it implies that the coordinate system itself has rotated anticlockwise by the same angle. This relationship highlights the concepts of active and passive transformations in coordinate systems. Understanding these transformations is crucial for applications in physics and engineering.
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If we change the orientation of a coordinate system as shown above, (the standard eluer angles , ##x_1y_1z_1## the initial configuration and ##x_by _b z_b## the final one), then the formula for the coordinates of a vector in the new system is given by

##x'=Ax##
where ##A=\left[\begin{array}{ccc}\cos \psi \cos \phi-\cos \theta \sin \phi \sin \psi & \cos \psi \sin \phi+\cos \theta \cos \phi \sin \psi & \sin \psi \sin \theta \\ -\sin \psi \cos \phi-\cos \theta \sin \phi \cos \psi & -\sin \psi \sin \phi+\cos \theta \cos \phi \cos \psi & \cos \psi \cdot \sin \theta \\ \sin \theta \sin \phi & -\sin \theta \cos \phi & \cos \theta\end{array}\right]##

We observe that the matrix is orthogonal with determinant ##+1## so it's a rotation matrix.
So its effect is to rotate a vector,hence ##x'## will be nothing but ##x## rotated.

So if ##A## represents a clockwise rotation of vectors by ##α## cannot I say that actually the coordinate system has rotated anticlockwise by the same amount ##α##?
 
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Filip Larsen said:
In short, yes. For a bit more details search for the concept of active and passive transformations.
thank you so much :) .
 
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