Express Newton's law of gravitation in vector form. Derive Newton's third law of motion.

Explanation

Consider two particles A and B of masses `M_(1) and M_(2)` and separated by distance r.

The mass `M_(2)` is attracted by `M_(1)` with a force `vec(F)_(12)` and is given by
`vec(F)_(12)= - (GM_(1)M_(2))/(r^(2)) hat(r )_(21)` ..(i)
where `hat(r )_(21)` is unit vector from B to A. Negative sign shows the attractive nature of force. The mass `M_(1)` is attracted by `M_(2)` with a force `vec(F)_(12)` and is given by
`vec(F)_(21)= (GM_(1)M_(2))/(r^(2)) hat(r )_(12)` ..(ii)
where `hat(r )_(12)` is unit vector from A to B
But `hat(r )_(12)= - hat(r )_(21)`
`therefore vec(F )_(21)= (GM_(1) M_(2))/(r^(2)) hat(r )_(21)` ...(iii)
From (i) and (iii) `vec(F)_(12)= - vec(F)_(21)` which is Newton.s third law of motion. So gravitational forces between two bodies are equal in magnitude, but opposite in direction.