The gravitational vector field \(\vec{g}(\vec{r})\) is defined as the gravitational force exerted on a small object located at a particular position \(\vec{r}\) divided by that object's mass.

(a) Sketch the gravitational vector field produced by Earth at a sampling of positions around Earth. Make sure your vector lengths indicate the correct relative sizes.

(b) Fxpress the gravitational force exerted by Earth on an object of mass \(m_{\text {obj }}\) located a distance \(r_{\text {obi }}\left(>r_{E}\right)\) from Earth's center in terms of Earth's gravitational vector field.