Newton's Law of Universal Gravitation - Vector Form

Vector Form

Newton's law of universal gravitation can be written as a vector equation to account for the direction of the gravitational force as well as its magnitude. In this formula, quantities in bold represent vectors.

\mathbf{F}_{12} = - G {m_1 m_2 \over {\vert \mathbf{r}_{12} \vert}^2} \, \mathbf{\hat{r}}_{12}

where

F12 is the force applied on object 2 due to object 1,
G is the gravitational constant,
m1 and m2 are respectively the masses of objects 1 and 2,
|r12| = |r2 − r1| is the distance between objects 1 and 2, and
is the unit vector from object 1 to 2.

It can be seen that the vector form of the equation is the same as the scalar form given earlier, except that F is now a vector quantity, and the right hand side is multiplied by the appropriate unit vector. Also, it can be seen that F12 = −F21.

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