Find angle and axis of rotation

[EvLer]

find angle and axis of rotation that sends [x1,x2,x3] to [x2,x3,x1]
I am not sure where to begin with this.

 

[arildno]

1. What unit vector is perpendicular to BOTH of your given vectors?
2. What is the angle between your vectors?

 

[AlephZero]

Think about special cases. For example, if a point is on the axis of rotation, then it is the same before and after the rotation.

Also, think about whether repeating the rotation several times will bring the point back to its original position.

 

[EvLer]

1. What unit vector is perpendicular to BOTH of your given vectors?
2. What is the angle between your vectors?

1. I am not sure, I was trying to come up with a vector whose dot product with each of the given ones would give a zero, but i can't.
I just can't visualize this, although I do see that that vector would be the the axis of rotation and then the angle of rotation I can find by cos(theta) = (a(dot)b)/(|a|*|b|)
how should I approach finding the orthogonal vector?
thanks again

 

[arildno]

"I just can't visualize this, although I do see that that vector would be the the axis of rotation and then the angle of rotation I can find by cos(theta) = (a(dot)b)/(|a|*|b|)"
This is, indeed, the answer to 2.!

Now, have you learned about the cross product of vectors?

 

[EvLer]

Now, have you learned about the cross product of vectors?

yeah, back in calculus... oh, I see, I just looked it up... thanks.
But the thing is I am in Linear algebra course and we are on the topic of rotations, reflections as linear transformations, so I am wondering if I am supposed to see something from that view rather than from calculus.
Any input on that?

 

[AlephZero]

If you want to find the rotation matrix R such that x' = Rx, you can take x to be the 3 vectors [1 0 0] [0 1 0] [0 0 1].

That will give you an matrix equation X' = RX where X = I, and so R = X'. Then interpret R as a direction and an angle.

 

[EvLer]

yeah exactly, how DO i interpret it as an angle and an axis? I can find the matrix no problem, but interpreting it is where i am not sure.

 

[HallsofIvy]

Any rotation matrix has one real eigenvalue, 1, and two complex eigenvalues, $cos(\theta)\pm isin(\theta)$. In this case the eigenvalues are the three "cube roots of unity", 1 and $\frac{1}{2}\pm i\frac{\sqrt{3}}{2}$. The complex roots correspond to $\theta= \frac{2\pi}{3}$. Any eigenvector corresponding to eigenvalue 1 must satisfy y= x, z= y, and x= z. In other words, the axis is the line z= y= x.

 

[AlephZero]

You can answer this problem, knowing anything about vectors or matrices. Just look at what you are given and think about what it means.

Any point on the axis of rotation doesn't move when it is rotated. If [x1 x2 x3] rotates to [x2 x3 x1], then any point [a a a] is rotated to [a a a].

Since the rotation is just a permutation of the 3 coordinate values, three rotations get you back to where you started from. So the angle of rotation must be 2 pi/3.

HallsofIvy's method is more general, of course - but why do it the hard way when there's an easy way?

 

[HallsofIvy]

Very nice!

 

[sgrandhi]

You can answer this problem, knowing anything about vectors or matrices. Just look at what you are given and think about what it means.

Any point on the axis of rotation doesn't move when it is rotated. If [x1 x2 x3] rotates to [x2 x3 x1], then any point [a a a] is rotated to [a a a].

Since the rotation is just a permutation of the 3 coordinate values, three rotations get you back to where you started from. So the angle of rotation must be 2 pi/3.

HallsofIvy's method is more general, of course - but why do it the hard way when there's an easy way?
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This is not the right way to go about it. The angle of rotation is formally calculated from the dot product. The angle of rotation is not 2*pi/3. It is true however, that 3 such rotations brings you back to the vector you started with. The 3 rotations are equal. But the complete cycle of 3 rotations does not correspond to 2*pi since the axis of rotation is different in each case.

 

[sgrandhi]

For example, try [x1 x2 x3] = [1 0 0]; => [x2 x3 x1] = [0 0 1]. i.e the x and z axes. The axis of rotation is the y-axis and the angle is pi/2. (From the dot product)

For the next rotation in the cycle, z axis to y axis, the axis of rotation is the x-axis and the angle is pi/2 again. Then the y-axis to the x-axis again, the axis of rotation is z and the angle is pi/2. So it is not necessary the angles of rotation add up to 2*pi, since the axes of rotation are different in each case.

 

[Kasiopeja]

Hello,

Maybe someone know about sun rotation? How long people are watching sun rotation. For me the top question is - whether the speed of rotation is changing?