Is there a way of subtracting two vectors in spherical coordinate system without first having to convert them to cartesian or other forms?
Since I have already searched and found the difference between Two Vectors in Spherical Coordinates as,
$$|\vec{x}-\vec{x'}|=(\rho^{2}+\rho'^{2}-2\rho\rho'[\cos(\theta-\theta')+\sin\theta\sin\theta'(cos(\phi-\phi')-1))])^{\frac{1}{2}}$$
which I believe the radius of displaced vector. I still didn't get any way to find the theta (angle from positive z axis)and psi(angle from positive x axis). Please help.
The given expression is the distance between two points whose spherical coordinates are given. It is derived on Cartesian coordinate basis considering differences of $r,\theta,\phi$ which is the most convenient method.
If $\rho,\rho^{'},\phi, \phi^{'}, \theta $ are given two values of $ \theta ^{'}$ can be calculated.