How Can I Visualize 3D Rotation Transformations in Linear Algebra?

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SUMMARY

This discussion focuses on visualizing 3D rotation transformations in linear algebra, specifically for classical mechanics applications. The user successfully constructs transformation matrices for two specific rotations: one that rotates the z-axis 45 degrees toward the x-axis around the y-axis, and another that rotates a coordinate system 120 degrees about an axis making equal angles with the original axes. The matrices provided are: 1) 

1   0       0
0   1       0
√2/2  -√2/2  √2/2
and 
-0.5   √3/2   0.5
0.5   -0.5   √3/2
√3/2   0.5   -0.5
Images linked in the discussion assist in visualizing these transformations.

PREREQUISITES
  • Understanding of transformation matrices in linear algebra
  • Familiarity with 3D coordinate systems
  • Knowledge of rotation transformations
  • Basic skills in visualizing geometric transformations
NEXT STEPS
  • Study the derivation of rotation matrices in 3D space
  • Learn about Euler angles and their application in 3D rotations
  • Explore software tools like MATLAB or Python's Matplotlib for visualizing 3D transformations
  • Investigate the implications of rotation transformations in classical mechanics
USEFUL FOR

Students in classical mechanics, linear algebra enthusiasts, and anyone interested in visualizing and applying 3D rotation transformations in mathematical and physical contexts.

don_anon25
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These problems are actually for my classical mechanics class, but they are linear-algebra based. I can construct a transformation matrix, but I have trouble visualizing the rotations, particularly in 3-space. So if someone could help me get a pictorial idea of what's actually happening, then the problems would be much easier!

1) Find the transformation matrix that rotates the x3 (z) axis of a regular coordinate system 45 degrees toward x1 (x) around the x2 (y) -axis.
Here's the matrix I got for this one:
1 0 0
0 1 0
sqrt2/2 -sqrt2/2 sqrt2/2


2) Find the transformation matrix that rotates a rectangular coordinate system through an angle of 120 degrees about an axis making equal angles with the original three coordinate axes.
Here's the matrix I came up with for this one:
-.5 sqrt3/2 .5
.5 -.5 sqrt3/2
sqrt3/2 .5 -.5

Thanks!
 
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A couple images are posted here:

http://www.akiti.ca/RotateTrans.html

They are not exactly the transformation you have described but they should help you picture what is going on. And you should be able to easily modify the axes for your own application (I don't think your transformation matrix is correct).

Regards,


Duncan
 

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