Confusion related to system of coordinates....

Hello,
I am a little confused about the system of coordinates the spherical, cartesian and cylindrical.
Taking for example the spherical and cartesian coordinates. Suppose I have a vector at point (x1, y1, z1) in the cartesian system and the vector is Ax i + Ay j + Az k
Now I want to transform it to spherical coordinates. The equations for space transformation from Cartesian to Spherical are:

r = √(x² + y² + z²) ; θ = arctan (√(x² + y²) / z) ; Φ = arctan(y/x)

So I think I can use x1, y1, z1 in these equations to find the point in spherical coordinates where the vector is located. Now how do I transform the vector to spherical notation? Do I use the same equations? Or do I substitute these equations for the cartesian unit vectors in Ax i + Ay j + Az k and get the spherical counterpart
The unit vectors in cartesian coordinates are related to the unit vectors of spherical coordinates by the equations:

i = sinθ cosΦ r + cosθ cosΦ θ - sinΦ Φ
j = sinθ sinΦ r + cosθ sinΦ θ + cosΦ Φ
k = cosθ r - sinθ θ

So do I use these and substitute in place of the cartesian unit vectors in the A vector and get my spherical vector?
Or the 3rd possibility is that I transform Ax, Ay, Az from the 1st set of transformation equations and transform the cartesian unit vectors by the second set of equations, and substitute all that in Ax i + Ay j + Az k and get the vector in spherical coordinates?

spherical coordinate system have spherical unit vectors. If you want to write down your vector in spherical coordinate system, you will need to find the each component in terms of spherical unit vectors.

The r θ Φ in the second set of equations are the unit vectors. I found the answer to the above query and we do have to transform Ax Ay Az according to the 1st set of equations and the unit vectors according to the second set keeping in mind that here θ Φ are the angles set by x1, y1, z1 and then putting everything in the vector Ax i + Ay j + Az k to get the spherical coordinate vector.

First set of formulas is suficient.
First translate your vector start coordinate.
That is segment (x1-0,..) into spherical or whatever else system you need. Than work in coordinate system with sentrum in (X1,...).
Than obviously your vector coordinates will be translation of (X-X1,..) in your new system. Now you have to shift your new system back to centrum 0.
In spherical system it is quite easy to do. (angles just added...etc.).
Basically that is the reason why physicists prefer to work in spherical system.
Forces. etc. can be easily represented and calculated geometrically.
Things become visible.
Don't forget about signes!

use the 3rd method. but just transform the AX AY AZ and then write the unit vectors of spherical system with them.