Am I doing this right? Doesn't feel right. (find axis of rotation)

In summary, the homework statement asks for the parametric equation for the axis of rotation in a matrix A given the two standard rotations. The Attempt at a Solution provides the correct answer of [0.7174, -0.4142, 1]'.
  • #1
skyturnred
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0

Homework Statement



So here is the question:

Matrix A corresponds to the linear transformation T obtained by first rotating a vector in R3 through angle ∏/3 about the z axis and then through angle ∏/4 about the x-axis. Find the parametric equation for the axis of rotation.

Homework Equations





The Attempt at a Solution



Finding matrix A: First I write down the two standard rotations with the first one on the right and multiply them:

5AEMj.jpg


This gives me matrix A. I then take the result and subtract the 3x3 identity matrix (so Mat(A) - I3). I augment this by the 3x1 zero vector and rref. So the following is what I am rref-ing. (so I am solving this system (A-I)[w]=0, and the axis parallel to [w] is the axis of rotation)

CQBb0.jpg


But when I rref this, I get the following:

W3=t where t is in the reals
W2=-0.4142t
W1=0.7174t

This doesn't seem right to me.. so the parametric form of the axis of rotation is this:

x=0.7174t
y=-0.4142t
z=t

Thanks so much in advance!
 
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  • #2
The rotation axis must be perpendicular to both the starting vector and the ending vector, how do you find such a vector that is perpendicular to both?
 
  • #3
I don't quite understand, what are the "starting" and "ending" vectors? Does that mean that my method above is wrong?

I know that the cross product is how you find a vector that is perpendicular to two other vectors.
 
  • #4
I'm sorry for providing wrong information. Please ignore my last post.
Regarding the problem. If you wrote down the rotation matrix correctly, you should get the right answer. The rotation axis is nothing but the eigenvector of the rotation matrix with eigenvalue 1, which, after I solved for it, is exactly [0.7174, -0.4142, 1]'. If you get a different answer, try to check your calculation. FYI, the rotation matrix I got was
[0.5,-0.866,0
0.6124,0.3536,-0.7071
0.6124,0.3536,0.7071]
Check your work.
 
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FAQ: Am I doing this right? Doesn't feel right. (find axis of rotation)

What is the purpose of finding the axis of rotation?

The axis of rotation is used to determine the direction and magnitude of the rotational movement of an object. It is an important concept in rotational mechanics and can help in analyzing the motion of objects.

How do I find the axis of rotation?

The axis of rotation can be found by identifying the axis around which the object is rotating. This can be done by looking at the direction of the rotational movement or by using mathematical equations to determine the axis.

What if I can't find the axis of rotation?

If you are having trouble finding the axis of rotation, try to visualize the direction of the rotational movement and determine the axis based on that. You can also seek help from a teacher or a more experienced scientist.

How does the axis of rotation affect the motion of an object?

The axis of rotation determines the direction and magnitude of the rotational movement of an object. It is an important factor in understanding the motion of objects, as it can affect the speed, acceleration, and other properties of the rotation.

Can the axis of rotation change?

Yes, the axis of rotation can change depending on the type of motion or external forces acting on the object. For example, if a force is applied at a different angle, the axis of rotation may shift or change direction.

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