Rotation of a vector along two axes (of which one is angle-dependent)

In summary, the conversation revolves around finding an expression for a unit vector in the direction of F. The speakers discuss the projections of F on the three axes and consider the influence of rotation on the vector. They also mention using spherical coordinate transformation to solve the problem, but note the need to add a negative sign. The final question asks for clarification on why the vector must be rotated by sin(alpha) to get the correct answer.
  • #1
Andrea94
21
8
1656545677417.png


I have been trying to determine an expression for a unit vector in the direction of F for hours now.
I think the expression is supposed to look something kind of like this,

1656545742697.png


But I don't understand at all how to arrive at this expression.
Any help?
 
Last edited:
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  • #2
If you relocate the real vector F to point O, keeping its direction, locating the tail exactly at O, what its projections on the three axes would be?
 
  • #3
Lnewqban said:
If you relocate the real vector F to point O, keeping its direction, locating the tail exactly at O, what its projections on the three axes would be?

On the z-axis it is clearly cos(beta) since that part of the rotation is not influenced by alpha. For the x-axis, I visualize that if alpha=0 then it is -sin(beta) and if alpha != 0 then this is the same as rotating -sin(beta) by cos(alpha). But I cannot figure out the y-axis.
 
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  • #4
Look down perpendicularly to the x-y plane.
 
  • #5
Lnewqban said:
Look down perpendicularly to the x-y plane.

So the only way we have a y-component is if beta != 0 AND alpha != 0, in which case the component along y from the beta part is sin(beta) (because this will be a diagonal vector contributing both to the y component and negatively to the x-component). So I can see the sin(beta) part, but I don't understand why I must rotate it by sin(alpha) (and not eg cos(alpha)) to get the correct answer.
 
  • #6
I think I get it based on spherical coordinate transformation,

1656595137581.png


The first column corresponds to my problem, but I have to add a negative sign because of the way my directions are set-up.
 
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  • #7
Andrea94 said:
So I can see the sin(beta) part, but I don't understand why I must rotate it by sin(alpha) (and not eg cos(alpha)) to get the correct answer.
May it be because it is a projection of one projection of the actual vector F?
 
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  • #8
Lnewqban said:
May it be because it is a projection of one projection of the actual vector F?
Yes specifically the components I am looking for 😌. Thanks for the help!
 
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1. What is the purpose of rotating a vector along two axes?

The purpose of rotating a vector along two axes is to change its orientation in a 3-dimensional space. This allows for the vector to be aligned with a specific direction or to be transformed into a new coordinate system.

2. How is the rotation of a vector along two axes calculated?

The rotation of a vector along two axes is calculated using a combination of trigonometric functions and matrix multiplication. The specific equations used depend on the rotation angles and axes involved.

3. Can a vector be rotated along two axes simultaneously?

Yes, a vector can be rotated along two axes simultaneously by composing the individual rotations into a single transformation matrix. This is known as a compound rotation.

4. What is the difference between rotating a vector along two axes and rotating it along one axis?

The main difference between rotating a vector along two axes and rotating it along one axis is the resulting orientation of the vector. Rotating along two axes allows for a greater range of orientations, while rotating along one axis is limited to a single plane of rotation.

5. Are there any real-world applications for rotating a vector along two axes?

Yes, there are many real-world applications for rotating a vector along two axes. Some examples include computer graphics, robotics, and navigation systems. Rotating a vector along two axes allows for precise control and manipulation of objects in 3-dimensional space.

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