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

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The discussion focuses on visualizing 3D rotation transformations in linear algebra, particularly for a classical mechanics class. The user has constructed transformation matrices for specific rotations but struggles with visualizing these rotations in three-dimensional space. Two transformation matrices are presented: one for a 45-degree rotation around the y-axis and another for a 120-degree rotation about an axis making equal angles with the coordinate axes. A link to images that may aid in understanding these transformations is provided, along with a suggestion that the user's transformation matrix may be incorrect. Visualization tools and resources are emphasized as essential for grasping the concepts of 3D rotations.
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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