# Vector form of Newton’s Law of Gravitation

## Vector form of Newton’s Law of Gravitation

Newton’s Law of Gravitation states that everybody in this universe attracts every other body with a force which is directly proportional to the product of their masses and is inversely proportional to the square of the distance between them.

Consider two points mass bodies A and B of masses m₁ and m₂ placed distance r apart.

Let,

$$\overrightarrow{{{r}_{12}}}$$ = Unit vector from A to B.

$$\overrightarrow{{{r}_{21}}}$$ = Unit vector from B toA.

$$\overrightarrow{{{F}_{21}}}$$ = Gravitational force exerted on body A by body B.

$$\overrightarrow{{{F}_{21}}}$$ = Gravitational force exerted on body B by body A.

According to Newton’s Law of Gravitation:

$$\overrightarrow{{{F}_{12}}}=-G\frac{{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\widehat{{{r}_{21}}}$$.

Here, negative sign indicates that the direction of $$\overrightarrow{{{F}_{12}}}$$ is opposite to that of $$\overrightarrow{{{F}_{21}}}$$. In fact, the negative sign shows that the gravitational force is attractive in nature.

As, $$r=|\overrightarrow{{{r}_{12}}}|=|\overrightarrow{{{r}_{21}}}|$$.

So,

$$\overrightarrow{{{F}_{12}}}=-\frac{G{{m}_{1}}{{m}_{2}}}{|\overrightarrow{{{r}_{21}}}{{|}^{2}}}\widehat{{{r}_{21}}}$$ … (1)

Similarly, $$\overrightarrow{{{F}_{12}}}=-\frac{G{{m}_{1}}{{m}_{2}}}{|\overrightarrow{{{r}_{12}}}{{|}^{2}}}\widehat{{{r}_{12}}}=\frac{G{{m}_{1}}{{m}_{2}}}{|\overrightarrow{{{r}_{12}}}{{|}^{2}}}\widehat{{{r}_{21}}}$$ … (2) $$\left[ \because \,\,\widehat{{{r}_{12}}}=-\widehat{{{r}_{21}}} \right]$$.

From equations, (1) and (2), we have:

$$\overrightarrow{{{F}_{21}}}=-\overrightarrow{{{F}_{12}}}$$. If $$\overrightarrow{{{r}_{1}}}$$ and $$\overrightarrow{{{r}_{2}}}$$ are the position vectors of point masses m₁ and m₂ at points A and B, as shown in below figure. Then,

$$\overrightarrow{{{r}_{12}}}=\overrightarrow{{{r}_{2}}}-\overrightarrow{{{r}_{1}}}$$.

Now,

$$\overrightarrow{{{F}_{21}}}=-\frac{G{{m}_{1}}{{m}_{2}}}{{{r}^{3}}}\widehat{{{r}_{12}}}=-\frac{G{{m}_{1}}{{m}_{2}}}{|\overrightarrow{{{r}_{12}}}{{|}^{3}}}\widehat{{{r}_{12}}}$$.

$$\overrightarrow{{{F}_{21}}}=-\frac{G{{m}_{1}}{{m}_{2}}}{|\overrightarrow{{{r}_{2}}}-\overrightarrow{{{r}_{1}}}{{|}^{3}}}\left( \overrightarrow{{{r}_{2}}}-\overrightarrow{{{r}_{1}}} \right)$$.