Given B= (10/r)i+( rcos θ) j+k in spherical coordinates. Find Cartesian points at (-3,4,0)

The question involves converting a vector given in spherical coordinates to Cartesian coordinates and then evaluating it at a specific Cartesian point. However, there seems to be a misunderstanding in how the question is posed. Let’s clarify the concepts involved before directly addressing the problem as it’s presented.

1. Clarification on Coordinates and Vector Fields:

– Spherical coordinates are typically denoted by ((rho, theta, phi)), where (rho) is the radial distance, (theta) is the azimuthal angle, and (phi) is the polar angle.

– Cartesian coordinates are denoted by ((x, y, z)).

– The vector field given, (mathbf{B} = left(frac{10}{r}mathbf{i} + (r costheta) mathbf{j} + mathbf{k} right)), appears to be defined with an assumption of some transformations that aren’t standard for spherical to Cartesian conversion. Specifically, (r) in the context given seems to represent a position vector magnitude, but without clearer definition, its conversion context is ambiguous. In standard spherical coordinates, (r = rho), the radial distance from the origin.

2. Conversion Process:

To approach the instruction literally, converting a vector from spherical to Cartesian coordinates generally involves using transformation equations based on the definitions of the spherical coordinates:

– (x = rho sinphi