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

AI Thread Summary
The discussion focuses on deriving an expression for a unit vector in the direction of a vector F, particularly when considering rotations around two axes, alpha and beta. The z-axis component is identified as cos(beta), while the x-axis component is linked to -sin(beta) adjusted by cos(alpha) when alpha is not zero. The y-axis component arises only when both angles are non-zero, with sin(beta) contributing to y and negatively affecting x. The confusion lies in why the y-component must be rotated by sin(alpha) rather than cos(alpha), which is clarified through the concept of projections in spherical coordinates. Overall, the participants seek to understand the correct expressions for the vector's components under these rotational transformations.
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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