# Why should the position vector be noted as $R\hat{R}$ in spherical polar coordinates?

Why should the position vector be noted as $$R\hat{R}$$ in spherical polar coordinates? Now i did the calculation like this: $$\vec R = R \sin\theta \cos\phi \hat{i} + R \sin\theta \sin\phi \hat{j} + R \cos\theta \hat{k}$$ so now i am manipulating the unit vectors. As :- $$\hat{R}= \frac{\frac{\partial \vec{R}}{\partial R}}{\left|\frac{\partial \vec{R}}{\partial R}\right|}=\sin\theta \cos\phi \hat{i} + \sin\theta \sin\phi \hat{j} + \cos\theta \hat{k}$$ by doing similiar calculations i found $$\hat{\theta}=\cos\theta \cos\phi \hat{i} + \cos\theta \sin\phi \hat{j} -\sin\theta\hat{k}$$. Similarly I found $$\hat{\phi}= \cos\phi \hat{i} + \sin\phi\hat{j}$$ now position vector can be written as $$\vec R= [\vec R. \hat{R}]\hat{\theta} + [\vec R. \hat{\theta}]\hat{\theta} + [\vec{R},\hat{\phi}] \hat{\phi}$$. Which gives me $$\vec{R} = R\hat{R} + R\sin\theta \hat{\phi}$$ not $$R\hat{R}$$ now where i am misunderstanding or miscalculating ?

You made a mistake when calculating $$\hat{\phi}$$. We have
$$\frac{d\hat{R}}{d\phi} = -R\sin\theta\sin\phi \hat{i} + R\sin\theta\cos\phi \hat{j}$$ so $$\hat\phi = \frac{\frac{d\hat{R}}{d\phi}}{\left|\frac{d\hat{R}}{d\phi}\right|} = \frac{ -R\sin\theta\sin\phi \hat{i} + R\sin\theta\cos\phi \hat{j}}{R\sin\theta}= - \sin\phi\hat{i} + \cos\phi\hat{j}$$
Now we have $$\left\langle \vec{R}, \hat\phi\right\rangle = -R\sin\theta\cos\phi\sin\phi + R\sin\theta\sin\phi\cos\phi = 0$$ which gives the correct result $$\vec{R} = R\hat{R}$$.