   Chapter 12.5, Problem 91E

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# Kepler's Laws In Exercises 87-94, you are asked to verify Kepler's Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector-valued function r. Let r = ‖ r ‖ , let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of the planet.Prove that d d t [ r r ] 1 r 3 [ ( r × r ' ) × r ] .

To determine

To prove: The relation ddt[rr]=1r3[(r×r')×r].

Explanation

Given:

The relation:

ddt[rr]=1r3[(r×r')×r]

Proof:

Consider the following expression:

ddt[rr]

ddt[rr]=rrr(drdt)r2=rr'r[(rr)/r]r2=r2r'(rr)rr3=(x2+y2+z2)(x'i+y'j+z'k)(xx'+yy'+zz')(xi+yj+zk)r3

Simplify further as follows:

ddt

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