Coordinate Transformations

[don_anon25]

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!

 

[DuncanM]

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

 

[tacman]

Hi there,

Sure, I'd be happy to help you visualize these rotations. In both cases, we are rotating the coordinate system in three-dimensional space. So to visualize this, let's start by drawing a 3D Cartesian coordinate system with the x, y, and z axes labeled. This will serve as our original coordinate system.

1) For the first problem, we are rotating the z-axis (x3) 45 degrees towards the x-axis (x1) around the y-axis (x2). To visualize this, imagine a point on the z-axis, say (0,0,1), which represents a unit vector in the z-direction. Now, as we rotate the coordinate system, this point will also rotate. The new position of this point after the rotation will be (sqrt(2)/2, 0, sqrt(2)/2), as you correctly calculated in your matrix. This corresponds to a vector that is 45 degrees from both the z and x axes. Similarly, any other point on the z-axis will also rotate 45 degrees towards the x-axis. This will result in a new coordinate system with the z-axis at an angle of 45 degrees from the original z-axis.

2) For the second problem, we are rotating the coordinate system through an angle of 120 degrees about an axis that makes equal angles with the original three axes. This means that the new axis will be at an angle of 60 degrees with each of the original axes. So to visualize this, imagine a point on the x-axis, say (1,0,0), which represents a unit vector in the x-direction. After the rotation, this point will now be at (0.5, sqrt(3)/2, 0.5), which corresponds to a vector that is 120 degrees from the original x-axis. Similarly, points on the y and z axes will also rotate 120 degrees. This will result in a new coordinate system where all three axes are rotated by 120 degrees from the original axes.

I hope this helps you visualize the rotations and make the problems easier for you. Good luck with your classical mechanics class!