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State Newton’s Law of gravitation expresses it in vector form.

Answer
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Hint: The Newton’s Law of gravitation is about the force of attraction that exists between all bodies in the universe. It tells us about how the force is dependent on certain parameters like mass, distance etc. The vector form is derived by expressing the vectors in a plane and then using their relations with each other.

Formula used:
 F = Gm1m2d2
F12 = -Gm1m2r2 r21

Complete step-by-step answer:
Newton’s Law of Gravitation is a universal law which states that every particle in the universe attracts other particles with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between the centers of the two particles.
Let a particle of mass m1 attract another particle of mass m2and the distance between the centers of both particles isdthen force of gravitation,Fbetween them is-
Fm1m2d2
On removing the proportionality sign, we get,

F = Gm1m2d2
Gis universal gravitational constant, it’s value is 6.67×1011 m3kg1s2 .
Let us consider two point masses A and B of mass m1 and m2 respectively. The distance between them isr.

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Here, r12 is the unit vector from A to B
r21 is the unit vector from B to A
F12 is the force of gravitation exerted by A on B
F21 is the force of gravitation exerted by B on A
By Newton’s Law of Gravitation, we know that,
F12 = Gm1m2r2 r12
F21 = Gm1m2r2 r21
As F12and F21are attractive forces, so their directions are opposite to each other. Therefore,
F12 =  F21
 F12 = Gm1m2r2 r21
As we know,
r = |r12| = |r21|
F12 = Gm1m2|r21|2 r21 - (1)
F12 = Gm1m2|r12|2 r12 = F12 = Gm1m2|r21|2 r21 [ r12 =  r21] - (2)
From eq (1) and eq (2), we have,
F12 =  F21

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 From the above figure,
r12 = r2  r1

r12 = r2  r1|r2  r1| - (3)
Substituting eq (3) in eq (1)

 F12 = Gm1m2|r2r1|3 (r2  r1)

To summarize, Newton’s Law of gravitation states that every particle in the universe attracts every other particle with the force such that, Fm1m2d2 and it’s vector form is F12 = Gm1m2|r2r1|3 (r2  r1)

Note: Gravitational forces on two given bodies are opposite in direction but equal in magnitude. Vectors are quantities which have magnitude as well as direction. The unit vector is given by r = r|r| .Gravitational forces are always attractive in nature and are one of the weakest forces.

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