Vector potential in spherical coordinates

[Another]

in this problem i can solve v = ω x r = <0, -ωrsinψ, 0> in cartesian coordinates

but i don't understand A in sphericle coordinates why?

(inside) A = ⅓μ₀Rσ(ω x r) = ⅓μ₀Rσωrsin(θ) θ^{^}

how to convert coordinate ?

 

[kuruman]

Look at figure 5.46. Unit vector $\hat \phi$ is in the xy-plane, perpendicular to the plane defined by $\vec {r}'$ and the z-axis in the direction of increasing $\phi$. Make a drawing of it in the xy plane and find its x and y components. Hint: It has a positive y-component and a negative x-component.