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

Question

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 ?

Answer 1

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 $\vec{R} = R\hat{R}$.