Coulomb's Law in Vector Form

What this figure shows. Two point charges $q_1$ and $q_2$ located at points A and B respectively, joined by a straight line of length $r_{12}$ (the separation between them). Two force vectors are drawn: $\vec{F}_{21}$, the force ON $q_2$ due to $q_1$, drawn at B pointing along the line away from A, with the unit vector $\hat{r}_{21}$ (directed along AB, away from B) used to write $\vec{F}_{21}=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r_{21}^2}\hat{r}_{21}$; and $\vec{F}_{12}$, the force ON $q_1$ due to $q_2$, drawn at A pointing along the line away from B, with unit vector $\hat{r}_{12}$ (directed along BA, away from A). The figure shows $\hat{r}_{12}$ and $\hat{r}_{21}$ pointing in exactly opposite direct …

Worked out. For two protons ($q_1=q_2=+1.6\times10^{-19}$ C, mass $1.67\times10^{-27}$ kg each) separated by $r=10^{-15}$ m, the electrostatic force is $F_e=\frac{1}{4\pi\epsilon_0}\frac{q^2}{r^2}=9\times10^9\times\frac{(1.6\times10^{-19})^2}{(10^{-15})^2}\approx2.3\times10^2$ N, while the gravitational force is $F_g=G\frac{m^2}{r^2}=6.674\times10^{-11}\times\frac{(1.67\times10^{-27})^2}{(10^{-15})^2}\approx1.86\times10^{-34}$ N, giving a ratio $F_e/F_g\approx1.23\times10^{36}$ -- the electrostatic force is about 36 orders of magnitude stronger than gravity at the same separation. The example draws out both similarities (both obey the inverse-square law, both are central forces acting along the joining line) and differences (gravity is always attractive while the electric force can be attractive or repulsive; gra …