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Vector Nature of Force

The electrostatic force is a vector quantity, meaning it has both magnitude and direction, described by Coulomb's Law.
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The statement of the theorem

Consider two point charges q1q_1 and q2q_2 located at positions r⃗1\vec{r}_1 and r⃗2\vec{r}_2 in R3\mathbb{R}^3. The electrostatic potential energy UU associated with this interaction is a scalar field defined by U(r⃗12)=kq1q2∣r⃗12∣U(\vec{r}_{12}) = k \frac{q_1 q_2}{|\vec{r}_{12}|}, where r⃗12=r⃗2−r⃗1\vec{r}_{12} = \vec{r}_2 - \vec{r}_1 and kk is Coulomb's constant. The force F⃗12\vec{F}_{12} exerted by q1q_1 on q2q_2 is a conservative force, and thus its vector nature is rigorously defined by the negative gradient of the potential energy function: F⃗12=−∇U(r⃗12)\vec{F}_{12} = -\nabla U(\vec{r}_{12})\nIn Cartesian coordinates, this yields the vector expression:\nF⃗12=−kq1q2r⃗12∣r⃗12∣3\vec{F}_{12} = -k q_1 q_2 \frac{\vec{r}_{12}}{|\vec{r}_{12}|^3}