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?
$$\vec{F}_g=\frac{Gm_1m_2}{|\vec{r}_{ij}|^3} \vec{r}_{ij}=\frac{Gm_1m_2}{|\vec{r}_{ij}|^3} |r_{ij}|\hat{r}_{ij}=\frac{Gm_1m_2}{|\vec{r}_{ij}|^2} \hat{r}_{ij}.$$
It's just one way textbooks write it, and is exactly equivalent to the right-most expression, which is probably the most obvious way to write the gravitational force in vector notation.
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 $$ F_i = \frac{GMm}{r^2}\hat r_i = F_g\hat r_i. $$
