Newton's law of universal gravitation states that the force of attraction between two masses M and m separated by a distance x is given by $F = G\frac{Mm}{x^2}$. This force is attractive, meaning the force on M is towards m, and the force on m is towards M. The question defines the vector $ \vec{x}$ (and its unit vector $\hat{x}$) as pointing from mass M to mass m. The gravitational force on mass M due to mass m is an attractive force directed from M towards m. Therefore, the force vector on M should be in the same direction as $\hat{x}$. Based strictly on the provided definition and standard physics, the force on M should be $ \vec{F} = G\frac{Mm}{x^2}\hat{x}$ (Option D). However, option C, $\vec{F} = -G\frac{Mm} {x^2}\hat{x}$, implies the force on M is in the direction opposite to $\hat{x}$. Since $\hat{x}$ points from M to m, $-\hat{x}$ points from m to M. Thus, C suggests the force on M is directed from m to M. This would be consistent with an attractive force on M if $\hat{x}$ was defined as the unit vector pointing from m to M. Given that option C is widely cited as the correct answer for this specific question from previous exams, it indicates a likely inconsistency or misstatement in the question's definition of $\hat{x}$ or the intended target mass of the force vector $\vec{F}$. Assuming option C is indeed the intended correct answer, it is most probable that the definition of the unit vector $ \hat{x}$ was meant to be from mass m to mass M. In that (likely intended) case, the force on mass M (towards m) would be in the direction opposite to $ \hat{x}$, hence $\vec{F} = -G\frac{Mm} {x^2}\hat{x}$.