Question

Why is r cubed in vector notation of universal law of gravitation?

Answer 1

In vector notation Newtonian force of gravity is
F⃗ =GMmr⃗ r3,F→=GMmr→r3,
where r=(x1−x′1)2+(x2−x′2)2+(x3−x′3)2−−−−−−−−−−−−−−−−−−√r=(x1−x1′)2+(x2−x2′)2+(x3−x3′)2and the radial vector r⃗ =x⃗ −x⃗ ′r→=x→−x→′. we can consider the unit vector r^=r⃗ rr^=r→r We can the write the vector notation as
F⃗ =GMmr2r^=Fgr^.F→=GMmr2r^=Fgr^.
which uses the scalar form of Newton's law of gravity. To write in component notation with r^=(r^1,r^2,r^3)r^=(r^1,r^2,r^3) we have
Fi=GMmr2r^i=Fgr^i.