Doubt in polar components of a vector

Ozma

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Joined
Oct 14, 2020
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I'm having troubles in understanding how my physics textbook finds the polar components of the vector \(\displaystyle \vec{p}\) (I've uploaded two images): by book introduces two orthogonal vectors \(\displaystyle \hat{u}_r\) and \(\displaystyle \hat{u}_\theta\) and says that \(\displaystyle \vec{p}=p \cos \theta \hat{u}_r-p \sin \theta \hat{u}_\theta\).
If I'm not wrong, the components should be the projections of \(\displaystyle \vec{p}\) onto \(\displaystyle \hat{u}_r\) and \(\displaystyle \hat{u}_\theta\), that is \(\displaystyle \vec{p}=p_r \hat{u}_r+p_\theta \hat{u}_\theta\) (in red and green in the second picture), but while I'm fine with the fact that the projection onto \(\displaystyle \hat{u}_r\) is \(\displaystyle p_r=p \cos \theta\) I don't understand the minus sign in the projection onto \(\displaystyle \hat{u}_\theta\) is \(\displaystyle p_\theta=-p \sin \theta\).
The main doubt is the fact that since by hypothesis \(\displaystyle \hat{u}_r\) and \(\displaystyle \hat{u}_\theta\) are orthogonal, the angle between \(\displaystyle \vec{p}\) and \(\displaystyle \hat{u}_\theta\) should be \(\displaystyle \theta+\pi/2\), so the angle between the extension of \(\displaystyle \vec{p}\) (the one dashed under \(\displaystyle \vec{p}\) in my drawing) and \(\displaystyle \hat{u}_\theta\) should be \(\displaystyle \pi-\pi/2-\theta=\pi/2-\theta\) and so \(\displaystyle p_\theta=p \cos (\pi/2-\theta)=p \sin \theta\); so I don't get the minus sign. Where am I doing a mistake?
 

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skeeter

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I don't understand the minus sign in the projection onto \(\displaystyle \hat{u}_\theta\) is \(\displaystyle p_θ=−p\sinθ\)
polar_vec.jpg
 
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