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Question: prove that the shortest path between two points on the...

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Prove that the shortest path between two points on the unit sphere is an arc of a great circle con- necting them. Hint: Without loss of generality, take one point to be (0,0, 1) and the other to be (sin uo . 0. cos uo). Let α (t) = (sin u (t ) cos u (t), sin u (t) sin v (t), cos u (t)), а t b, be an arbi- trary curve with u(a)-0, v(a)0, u(b) o, v(b)-0, calculate the arclength of a, and show that it is smallest when v()-0 for all t.) Prove that if P and Q are points on the unit sphere, then the shortest path between them has length arccos(P.2) a. b.
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