Rotating vectors on a unit sphere

In summary, the conversation is about rotating unit vectors on a unit sphere by 120 degrees around the origin. The initial vector has spherical coordinates (1, θ, ∅) and the goal is to obtain the coordinates of unit vectors that make 120 degrees with the given vector. The dot product relation is mentioned as a possible method, but it does not work for all values of theta and phi. The question is raised if this could be due to a subtlety in 3 dimensions. A solution is suggested, using a rotation matrix to take (0,0,1) to the given vector. The questioner also mentions knowing how to do this in 2 dimensions, but is unsure how to find the unit vectors in
  • #1
tut_einstein
31
0
Hi,

I want to rotate vectors through 120 and they are unit vectors so they lie on a unit spheres. So basically the tails of the vectors are at the origin and given one vector with spherical coordinates (1,θ,∅), how do I obtain the coordinates of the unit vectors that make 120 degrees with the given vector?

I tried using the dot product relation. But it doesn't seem to work for all values of theta and phi I pick for the initial one b/c sometimes, I get cosine and sine values that are greater than one.

Is it because I'm missing some kind of subtlety in 3 dimensions?

Thanks!
 
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  • #2
Anyone? I would really appreciate some help!
 
  • #3
Rotating them by 120 degrees around what axis?
 
  • #4
You want all the unit vectors (a circle's worth of them) that make a 120-degree angle with the given one? If your given vector is [itex](1,0,0)[/itex], then the unit vectors with a 120-degree angle to that are parameterized by [itex](-1/2,(\sqrt3/2)\cos\theta,(\sqrt3/2)\sin\theta)[/itex] for [itex]0\le\theta<2\pi[/itex].

If you have a different given vector, just multiply everything by any rotation matrix that takes [itex](0,0,1)[/itex] to the vector you were given.
 
  • #5
I need to rotate about the origin. I'm not sure what the axis is.

Also, tinyboss, I don't quite understand your answer. I know how to do it in 2 dimensions (when theta = pi/2, wheer theta is theta is the polar angle of spherical coordinates - angle made with the z-axis that is).

But when I move off the xy plane I don't know how to find the unit vectors that are 120 degrees apart from the given one.
 

Related to Rotating vectors on a unit sphere

1. What is a unit sphere?

A unit sphere is a mathematical concept that represents a perfect, symmetrical sphere with a radius of 1 unit. It is often used as a reference shape in various mathematical and scientific calculations.

2. How are vectors rotated on a unit sphere?

Vectors on a unit sphere can be rotated by using a combination of rotation matrices and trigonometric functions. The specific method may vary depending on the desired rotation and the notation being used.

3. What is the purpose of rotating vectors on a unit sphere?

Rotating vectors on a unit sphere can be used to model the movement of objects in physical space, such as the rotation of planets or the orientation of an aircraft in flight. It is also a useful tool in various mathematical and scientific applications.

4. What is the significance of unit vectors in this context?

Unit vectors are vectors with a magnitude of 1 unit, which are often used in conjunction with the unit sphere to represent rotations. They allow for a consistent and convenient way to describe and calculate rotations on a spherical surface.

5. Are there any real-world applications of rotating vectors on a unit sphere?

Yes, there are many real-world applications of rotating vectors on a unit sphere, such as in navigation systems, computer graphics, and robotics. They are also commonly used in physics and engineering to model and analyze the movement of objects in 3D space.

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