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

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Discussion Overview

The discussion revolves around determining an expression for a unit vector in the direction of a vector F, particularly focusing on its projections along two axes that are influenced by angles alpha and beta. The scope includes mathematical reasoning and conceptual clarification regarding vector projections and transformations in a three-dimensional space.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in deriving an expression for a unit vector in the direction of F and seeks assistance.
  • Another participant suggests relocating vector F to point O and questions what its projections on the three axes would be.
  • There is a discussion about the z-axis projection being cos(beta), while the x-axis projection is debated, with one participant visualizing it as -sin(beta) when alpha=0 and as a rotation of -sin(beta) by cos(alpha) when alpha is not zero.
  • A participant notes that a y-component exists only if both beta and alpha are non-zero, contributing to the y-component as sin(beta) and negatively to the x-component.
  • There is confusion regarding the necessity of rotating the y-component by sin(alpha) instead of cos(alpha) to achieve the correct answer.
  • One participant references spherical coordinate transformation and mentions needing to add a negative sign due to the setup of their directions.
  • Another participant reiterates the projection aspect of the vector F and seeks clarification on the reasoning behind the rotation by sin(alpha).

Areas of Agreement / Disagreement

Participants express differing views on the correct approach to determining the projections and the reasoning behind using sin(alpha) versus cos(alpha) for the y-component. The discussion remains unresolved with multiple competing viewpoints on the projections and transformations involved.

Contextual Notes

There are limitations regarding the assumptions made about the angles and the setup of the coordinate system, which may affect the interpretations of the projections. The discussion also highlights unresolved mathematical steps in deriving the correct expressions.

Andrea94
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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?
 
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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?
 
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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Look down perpendicularly to the x-y plane.
 
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.
 
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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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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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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