Square of the distance in three dimension space

Question

$|\mathbf r_2 - \mathbf r_1 |^3$ denotes square of the distance that will be the result of position of a point in three dimension space after carrying Pythagorean operation. But why it is cubed. Is it just symbolic of three dimension?

Answer 1

You can read it as $$ \frac{1}{|r_2-r_1|^2}\cdot\frac{r_2-r_1}{|r_2-r_1|} = \frac{\widehat{(r_2-r_1)}}{|r_2-r_1|^2}. $$ The first factor is the inverse square that you're used to, and the second is the normalized direction vector (meaning we divide by the length to make it have a length of 1).

See also https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation#Vector_form

Answer 2

It is cubed because we want a vector force :

So because the original gravitational is proportional to the inverse squared distance, putting the vector $r_2-r_1$ at the numerator allows to have vector force and compensating by the cubic norm to reach a total of homogeneous inverse squared distance.