Let P, Q be two points on the sphere of radius 1, centered at the origin....

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Let P, Q be two points on the sphere of radius 1, centered at the origin. Let L(t)=P+t(Q-P), with 0 ≤t≤ 1. If there exists a value of t in [0, 1] such that L(t) = 0, show that t=1, and that P = -Q. Let P, Q be two points on the sphere of radius 1. Assume that P-Q. Show that there exists a curve joining P and Q on the sphere of radius 1, centered at the origin. By this we mean there exists a curve C(t) such that C(t) =1, or if you wish ||C(t)|| = 1 for all t, and there are two numbers t₁ and t₂ such that C(t₁) = P and C(t₂) = Q. [Hint: Divide L(t) by its norm.] If P, Q are two unit vectors such that P = -Q, show that there exists a differentiable curve joining P and Q on the sphere of radius 1, centered at the origin. You may assume that there exists a unit vector A which is per- pendicular to P. Let P, Q be two points on the sphere of radius 1, centered at the origin. Let L(t)=P+t(Q-P), with 0 ≤t≤ 1. If there exists a value of t in [0, 1] such that L(t) = 0, show that t=1, and that P = -Q. Let P, Q be two points on the sphere of radius 1. Assume that P-Q. Show that there exists a curve joining P and Q on the sphere of radius 1, centered at the origin. By this we mean there exists a curve C(t) such that C(t) =1, or if you wish ||C(t)|| = 1 for all t, and there are two numbers t₁ and t₂ such that C(t₁) = P and C(t₂) = Q. [Hint: Divide L(t) by its norm.] If P, Q are two unit vectors such that P = -Q, show that there exists a differentiable curve joining P and Q on the sphere of radius 1, centered at the origin. You may assume that there exists a unit vector A which is per- pendicular to P.

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To prove each of the statements given 1 Let Lt P tQ P be defined as in the question If there exists ... View the full answer