Please help in transforming rectangle coordinates to spherical coordinates.

I am doing this as a practice of transforming using two approach by **a)** performing the cross product first and then transform from rectangle to spherical coordinates and **b)** Transform both to spherical coordinates first before performing the cross product. I notice that I don’t get the same answer. I checked carefully and find nothing wrong that I can tell. In b), it will give because being a position vector, the cross product is zero. Below is my work, please tell me what did I do wrong:

I start out in rectangle coordinates with letting:

**a)** Performing cross product first and transform:

Given equation of the component:

Same as

**b)** Transform both to spherical coordinates first before performing the cross product:

Using the same formula above, both with different .

As you can see, both only has the components only as expected from a position vector.

This imply you get a total different answer in two different procedures. It is my understanding you can perform the cross product either way but obviously it does not work. Please take a look and see what did I do wrong.

Thanks