Why the $r$ is cubed in the vector notation for of Newton's Law of Universal Gravitation? [duplicate]

I'm learning about astrodynamics on my own and I was wondering why the $r$ is cubed in the vector notation for of Newton's Law of Universal Gravitation:

$$\vec{F}_g=\frac{Gm_1m_2}{|\vec{r}|^3}\vec{r}$$

I am familiar with Newton's Law of Universal Gravitation of the form:

$$F_g=\frac{Gm_1m_2}{r^2}$$

Is there something obvious I'm missing?

• Explained in the answer to this question, and almost an exact duplicate of this question with "Coulomb" replaced by "Newton". Commented Jun 1, 2016 at 18:14

$$\vec{F}_g=\frac{Gm_1m_2}{|\vec{r}_{ij}|^3} \vec{r}_{ij}=\frac{Gm_1m_2}{|\vec{r}_{ij}|^3} |r_{ij}|\hat{r}_{ij}=\frac{Gm_1m_2}{|\vec{r}_{ij}|^2} \hat{r}_{ij}.$$
In vector notation Newtonian force of gravity is $$\vec F = \frac{GMm\vec r}{r^3},$$ where $r = \sqrt{(x_1 - x'_1)^2 + (x_2 - x'_2)^2 + (x_3 - x'_3)^2}$ and the radial vector $\vec r = \vec x - \vec x'$. we can consider the unit vector $\hat r = \frac{\vec r}{r}$ We can the write the vector notation as $$\vec F = \frac{GMm}{r^2}\hat r = F_g\hat r.$$ which uses the scalar form of Newton's law of gravity. To write in component notation with $\hat r = (\hat r_1, \hat r_2, \hat r_3)$ we have $$F_i = \frac{GMm}{r^2}\hat r_i = F_g\hat r_i.$$
• Instead of $x_1$, $x_2$, and $x_3$, I think you intended to write $x$, $y$, and $z$. Commented Aug 8, 2022 at 4:16