The problem here is that your vector is different for the points and . In the first case, it points in the direction and in the second it points in direction .
Can you give a reference for this problem? It doesn't look standard.
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:
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.