Spherical vectors and rotation of axes

In summary, the conversation discusses how to find the coordinates of a velocity vector in a new spherical coordinate system that arises from a rotation of the axes of the original coordinate system. This involves differentiating the components of a transformation matrix with respect to time and multiplying it by the column vector containing the velocities. The conversation also mentions using the law of cosines/sines to define the rotation angle and constructing a rotation matrix to multiply the velocities.
  • #1
meteo student
72
0
I have a velocity vector as a function of a latitude and longitude on the surface of a sphere. Let us assume I have a point V(lambda, phi) where V is the velocity. The north pole of this sphere is rotated and I have a new north pole and I have a point V'(lambda, phi) in the new system. I have the transformation matrix between the unrotated system and rotated system in terms of a position defined in terms of lambda and phi.
Rather than just multiplying the rotation matrix by this transformation matrix I must differentiate the components of the transformation matrix with respect to time and multiply that result by the column vector containing the velocities. Is that correct ?
 
Mathematics news on Phys.org
  • #2
What does this mean:
meteo student said:
The north pole of this sphere is rotated and I have a new north pole
Are you just wanting to find the coordinates of the velocity vector in a new spherical coordinate system that arises from a rotation of the axes of the original spherical coordinate system around a line through the origin of the original system?
 
  • #3
Yes exactly.

I have velocities in the unrotated frame with respect to a geographical north pole of the earth. I have also been given the position of a point in the unrotated frame as well the origin of the rotated north pole.

I have defined theta the rotation angle in terms of the coordinates of a position in the unrotated frame and rotated frame using law of cosines/sines. Now all I need to do is construct the rotation matrix and multiply the velocities ?
 
Last edited:

Related to Spherical vectors and rotation of axes

What are spherical vectors?

Spherical vectors are a type of vector that is defined in terms of its magnitude, direction, and angle relative to a fixed reference point. They are typically used to describe the position or movement of an object in 3-dimensional space.

How are spherical vectors represented mathematically?

Spherical vectors are often represented using spherical coordinates, which use the variables r (radius), θ (polar angle), and φ (azimuthal angle). These coordinates can be converted to Cartesian coordinates using trigonometric functions.

What is rotation of axes?

Rotation of axes is a mathematical operation that involves changing the orientation of a coordinate system in 3-dimensional space. It is often used to simplify the description of complex shapes or to make calculations easier.

How does rotation of axes affect spherical vectors?

Rotation of axes can change the coordinates of a spherical vector, but it does not change the magnitude, direction, or angle of the vector. This means that the vector will still represent the same position or movement in space, but its coordinates will be different.

What are some practical applications of spherical vectors and rotation of axes?

Spherical vectors and rotation of axes are used in a variety of fields, including physics, engineering, and computer graphics. They are often used to describe the motion of celestial bodies, the rotation of objects in 3-dimensional space, and the orientation of objects in virtual reality environments.

Similar threads

B Invariant under rotation: Banal, obvious, or noteworthy?
    • General Math
Replies
2
Views
1K
A Finding Minimal Mean Distance Curves on the Unit Sphere
    • General Math
Replies
2
Views
1K
B Defining handedness, right-left, or clockwise-counterclockwise
    • General Math
Replies
3
Views
1K
I Solving simultaneous vector equations
    • General Math
Replies
11
Views
1K
I Transforming Vector Fields between Cylindrical Coordinates
    • General Math
Replies
1
Views
816
I Velocity transformation between frames rotating relative to one another
    • Mechanics
Replies
1
Views
442
I Converting from spherical to cylindrical coordinates
    • General Math
Replies
13
Views
2K
Rotate Points on Sphere by Theta and Phi
    • MATLAB, Maple, Mathematica, LaTeX
Replies
10
Views
2K
A Rotation matrix and rotation of coordinate system
    • Mechanics
Replies
2
Views
993
A Converting this vector into polar form
    • General Math
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top