Rotation matrix and rotation of coordinate system

[Kashmir]

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 $α$?

 

[Filip Larsen]

In short, yes. For a bit more details search for the concept of active and passive transformations.

 

[Kashmir]

In short, yes. For a bit more details search for the concept of active and passive transformations.

thank you so much :) .