Rotation matrix and rotation of coordinate system

In summary, the formula for the coordinates of a vector in a new coordinate system is given by ##x' = Ax## where ##A## is an orthogonal matrix with a determinant of +1. This matrix represents a rotation of the vector, with ##x'## being the rotated version of ##x##. This means that if ##A## represents a clockwise rotation of vectors by ##α##, then the coordinate system has actually rotated anticlockwise by the same amount. This concept can be further explored through the concept of active and passive transformations.
  • #1
Kashmir
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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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  • #3
Filip Larsen said:
In short, yes. For a bit more details search for the concept of active and passive transformations.
thank you so much :) .
 

FAQ: Rotation matrix and rotation of coordinate system

1. What is a rotation matrix?

A rotation matrix is a mathematical tool used to represent the rotation of a coordinate system in three-dimensional space. It is a square matrix that describes the transformation of a set of coordinates from one coordinate system to another through rotation.

2. How is a rotation matrix calculated?

A rotation matrix is calculated using trigonometric functions such as sine and cosine. The values in the matrix are determined by the angle of rotation and the axis of rotation. The matrix can be calculated using various methods, such as the Euler angles method or the axis-angle method.

3. What is the purpose of a rotation matrix?

The main purpose of a rotation matrix is to describe the orientation of a coordinate system after a rotation has been applied. It is commonly used in fields such as computer graphics, robotics, and physics to represent the movement of objects in three-dimensional space.

4. How does a rotation matrix affect the coordinates of a point?

A rotation matrix affects the coordinates of a point by transforming them from the original coordinate system to a new one after a rotation. The new coordinates can be calculated by multiplying the rotation matrix with the original coordinates. This results in a new set of coordinates that represent the point in the rotated coordinate system.

5. Can a rotation matrix be used for rotations in any direction?

Yes, a rotation matrix can be used for rotations in any direction in three-dimensional space. The axis of rotation and the angle of rotation can be specified to rotate the coordinate system in the desired direction. This allows for a wide range of applications, from simple 2D rotations to complex 3D transformations.

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