can we find out the cross product of vectors in cylindrical or spherical system using determinant method like we do in cartesian system? thanks
No you can't: you need to convert the cylindrical or spherical co-ordinates to cartesian and then apply the determinant method. The three numbers in cylindrical or spherical co-ordinates have a completely different meaning to the cartesian values. You cannot combine them using any of the common vector algebra - including addition.
I'm a bit confused by your notation. You appear to have two coordinates that are angles one that is a distance, but that distance is labeled 'z'. I'm used to seeing cylindrical coordinates with one angle and two distances, one of them 'r' and one of them 'z, and spherical with two angles and a distance but with the distance being 'r'. Is your 'z' a radius, or a in the Cartesian 'z' direction. Or is this cylindrical coordinates with ρ and z being distances and phi being an angle? In either case, there would seem to be the issue that the unit vectors, while mutually perpendicular at any point in space, but they are not always in the same direction. For instance, r_hat at one point is, in general, not parallel to r_hat at some other point. Thus, your vectors wouldn't seem to even be defined unless you specify the local point they are referenced to -- and the origin doesn't have a well defined set of unit vectors, unless it is defined by convention.