I want to calculate dotproduct of $ e_r *e_\phi $ they are unit vectors in spherical. where my spherical coordinates is $(r,\phi,\theta)$
My attempt is first to convert them to cartesian:
which gives: after som algebra $sin \phi cos\phi(1-1)= 0$.
I was thinking that I could do this in spherical coordinates directly without converting to cartesian?
Because $e_r$=(1,0,0) , $e_\phi$=(0,1,0)
Like $ e_r *e_\phi $= (1,0,0) dot (0,1,0)=0 ?
The unit vectors $\hat r$, $\hat \theta$, and $\hat \phi$ are mutually orthogonal. To show explicitly that $\hat r$ and $\hat \phi$ are orthogonal, we take their inner product and observe that it is zero.
To that end we first write the spherical unit vectors in Cartesian coordinates as
$\hat r=\hat x\sin \theta \cos \phi+\hat y\sin \theta \sin \phi+\hat z \cos \theta$
and
$\hat \phi=-\hat x \sin \phi+\hat y\cos \phi$
Therefore, their inner product is given by
$$\begin{align} \hat r \cdot \hat \phi&=\sin \theta(-\sin \phi \cos \phi+\sin \phi \cos \phi)\\\\ &=0 \end{align}$$
as expected!