Exploring Rotation Matrices: Finite & Infinitesimal Rotations

[neelakash]

Homework Statement

Can anyone help me to proceed with this?

If we execute rotations of 90* about x-axis and 90* about y axis-what is the resulatant rotation matrix?Will the result commute if we rotate by changing the order?Will they commute if infinitesimal rotations are considered?

Homework Equations

The Attempt at a Solution

A qualitative thinking suggest that they will not commute for finite angle but may commute for infinitesimal angles...
...But I possibly cannot visualise the picture correctly.Can anyone refer to some picture?

Most importantly,how to proceed with the calculations?

By the way,what is the rotation matrix for rotation about x,y,z axes?

 

[Jhero]

Hello,

To answer your first question, the resultant rotation matrix for rotating 90* about the x-axis and then 90* about the y-axis would be:

R = [0 -1 0; 1 0 0; 0 0 1]

This can be derived by multiplying the individual rotation matrices for each axis:

Rx = [1 0 0; 0 0 -1; 0 1 0] and Ry = [0 0 1; 0 1 0; -1 0 0]

Rx*Ry = [0 -1 0; 1 0 0; 0 0 1]

To answer your second question, the order of rotation does matter in this case. If you change the order to rotating 90* about the y-axis and then 90* about the x-axis, the resultant rotation matrix would be:

R = [0 1 0; -1 0 0; 0 0 1]

This is not the same as the previous result and therefore, they do not commute.

For infinitesimal rotations, the result may or may not commute depending on the specific angles chosen.

To visualize the rotation, you can imagine a 3D coordinate system with a point at (1,0,0). After rotating 90* about the x-axis, the point would be at (1,0,-1). Then, rotating 90* about the y-axis would bring the point to (0,0,-1).

The rotation matrices for rotating about the x,y,z axes are:

Rx = [1 0 0; 0 cos(theta) -sin(theta); 0 sin(theta) cos(theta)]

Ry = [cos(theta) 0 sin(theta); 0 1 0; -sin(theta) 0 cos(theta)]

Rz = [cos(theta) -sin(theta) 0; sin(theta) cos(theta) 0; 0 0 1]

I hope this helps you with your calculations and understanding of rotations. Let me know if you have any further questions. Good luck!