How to form the transformation matrix for this

In summary, the conversation discusses the task of forming a transformation matrix that rotates the x1 axis of a rectangular coordinate system by 60 degrees towards the x2 and x3 axes. The speaker is unsure of how to interpret this instruction and asks for clarification. There are multiple ways to rotate the x1 axis in this manner, but the teacher ultimately clarifies that it should be rotated around the x3 axis.
  • #1
Byang
2
0
We were asked to form the transformation matrix that rotates the x1 axis of a rectangular coordinate system 60 degrees toward x2 and the x3 axis.
The thing is, I don't understand what it meant by rotating one axis toward the two other. Like, do I rotate x1 60 degrees toward the x2-x3 plane or does it mean something else? I tried to rotate x1 to the x2-x3 plane but I can never be sure of the angle between x1' and x2, and x1' and x3. How do I find the angles?
 
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  • #2
Byang said:
We were asked to form the transformation matrix that rotates the x1 axis of a rectangular coordinate system 60 degrees toward x2 and the x3 axis.
The thing is, I don't understand what it meant by rotating one axis toward the two other. Like, do I rotate x1 60 degrees toward the x2-x3 plane or does it mean something else? I tried to rotate x1 to the x2-x3 plane but I can never be sure of the angle between x1' and x2, and x1' and x3. How do I find the angles?

There are infinitely many possible ways of rotating the ##x_1## axis by ##60^o## towards the ##x_2-x_3## plane. For example, you could fix ##x_2 = 0## and rotate the ##x_1## axis towards the ##x_3## axis, or fix ##x_3 = 0## and rotate ##x_1## towards ##x_2##, etc. In all these cases you would be rotating ##x_1## by ##60^o## towards the ##x_2-x_3## plane. In general, you could fix an arbitrary point ##p = (0,p_2,p_3)## in the ##x_2-x_3## plane, and rotate the ##x_1## axis towards ##p##.

Basically, I don't know exactly what your question wants; perhaps they mean ##60^o ## towards ##x_2## and then ##60^o## towards ##x_3##, or maybe they mean two separate questions---one for an ##x_1 \to x_2## rotation and another for an ##x_1 \to x_3## rotation.
 
  • #3
Ray Vickson said:
Basically, I don't know exactly what your question wants
Haha, that's exactly my problem. But thank you for answering. We've cleared it up with our teacher and she said it was supposed to be "around x3 axis," which makes the problem a lot easier.
 
  • #4
Thread moved as it seems to be more of a general question than a specific homework problem.
 

1. How do I determine the size of the transformation matrix?

The size of the transformation matrix will depend on the number of dimensions in your original data and the desired number of dimensions in the transformed data. For example, if your original data has three dimensions and you want to transform it into two dimensions, your transformation matrix will be a 2x3 matrix.

2. What is the purpose of a transformation matrix?

A transformation matrix is used to transform data from one coordinate system to another. It can be used to rotate, scale, or translate data points in order to better visualize or analyze the data.

3. How do I fill in the values of the transformation matrix?

The values of the transformation matrix will depend on the specific transformation you want to apply to your data. For rotation, the values will correspond to the cosine and sine of the angle of rotation. For scaling, the values will correspond to the scaling factor for each dimension. It is important to determine the specific transformation you want to apply before filling in the values of the matrix.

4. Can I use a pre-made transformation matrix for my data?

It is possible to use a pre-made transformation matrix, but it may not be suitable for your specific data. It is important to understand the transformation you want to apply and the values needed for the matrix in order to accurately transform your data.

5. How do I apply the transformation matrix to my data?

To apply the transformation matrix, you will need to multiply it by your original data points. This will result in a new set of transformed data points. You can then plot or analyze these transformed data points to gain insight into your data in a different coordinate system.

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