I am using this coordinate system:
consider these two vectors in spherical coordinates:
$$\vec A=A_r \hat r +A_\phi \hat \phi + A_\theta \hat \theta= A_r\sin{\theta_A} \cos{\phi_A} \hat i +A_r\sin{\theta_A} \sin{\phi_A} \hat j +A_r\cos{\theta_A} \hat k$$ $$\vec B=B_r \hat r +B_\phi \hat \phi + B_\theta \hat \theta= B_r\sin{\theta_B} \cos{\phi_B} \hat i +B_r\sin{\theta_B} \sin{\phi_B} \hat j +B_r\cos{\theta_B} \hat k$$
In Cartesian coordinates, the formula of the cross product is:
$$\vec A \times \vec B= \begin{vmatrix} \hat i & \hat j & \hat k \\ A_x & A_y & A_z \\ B_x & B_y & B_z \\ \end{vmatrix} = \begin{vmatrix} \hat i & \hat j & \hat k \\ A_r\sin{\theta_A} \cos{\phi_A} & A_r\sin{\theta_A} \sin{\phi_A} & A_r\cos{\theta_A} \\ B_r\sin{\theta_B} \cos{\phi_B} & B_r\sin{\theta_B} \sin{\phi_B} & B_r\cos{\theta_B} \\ \end{vmatrix} $$
Which is equal to:
$$ \vec A \times \vec B = (A_yB_z-A_zB_y)\hat i + (A_zB_x-A_xB_z)\hat j + (A_xB_y-A_yB_x)\hat k $$
What is the formula of cross product in spherical coordinates?
In either form. matrix or in one line.