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 ?