Directional Notation for Forces and Gravity

Question

It is said that $F_{12}$ is the force on object 1 by object 2. In Newton's law of gravity, $$\vec{F_{12}} = -\frac{Gm_1m_2}{r_{12}^{2}}\hat{r}_{12}.$$ $\hat{r}_{12}$ points from object 1 to 2, $\vec{F_{12}}$ is the force on object 1 by object 2, and as gravity is attractive, this points from object 1 to 2 as well. So why is there a minus sign in Newton's law of gravity, considering these two vectors point in the same direction?

Please help me clear up my confusion.

Answer 1

Based on the way you have defined $\vec{F}_{12}$ and $\hat{r}_{12}$, the given equation is incorrect. The equation would be correct if $\vec{F}_{12}$ instead referred to the force produced by 1 acting on 2.