let △ABC be a triangle S2 (the two-dimensional sphere). Let the dual point C'∈S2 of C be defined by the following three conditions:(i) d(C' , A)=π/2(ii) d(C' , B)=π/2(iii) d(C' , C)≤π/2A' and B' are defined analogously. Thus we get a dual triangle △A' B' C'. More precisely, △ABC is a non-degenerate triangle, and it follows (you may assume) that △A' B' C' is too. (a) Show that the conditions (i)–(iii) determine a unique point C' by, in terms of vectors in ℝ3thinking. (For example, (i) is in terms of perpendicularity to interpret.)(b) Show that d(C' , X) = π/2 for all points X on the line (in the sentence of S2, so great circle) line(A, B).(c) Show that the dual triangle of the dual triangle is the original onetriangle is (△A'' B'' C''=△ABC) by proving that C''=C. (The othercases, A'' = A and B'' = B, are of course analogous.) I have attached an image of the questions incase the symbols dont show properly. If able please provide some explanation with the taken steps, thank you very much. let AABC be a triangle S². Let the dual point C'ES² of C be defined by the following three conditions:(i) d(C', A)=π/2(ii) d(C¹, B)=π/2(iii) d(C¹, C)≤π/2A' and B' are defined analogously. Thus we get a dual triangle AA' B'C'. More precisely, AABC is a non-degenerate triangle, and it follows (you may assume) that AA' B' C' is too.let AABC be a triangle S^2. Let the dual point C'ES^2 of C be defined by the following three conditions:(i) d(C', A)=π/2(ii) d(C¹, B)=π/2(iii) d(C¹, C)<π/2A' and B' are defined analogously. Thus we get a dual triangle AA' B' C'. More precisely, AABC is a non-degenerate triangle, andit follows (you may assume) that AA' B' C' is too.(a) Show that the conditions (i)-(iii) determine a unique point C' by, in terms of vectors in R³thinking. (For example, (i) is in terms of perpendicularity to interpret.)(b) Show that d(C', X) = π/2 for all points X on the line (in the sentence of S², so great circle) line(A, B).(c) Show that the dual triangle of the dual triangle is the original onetriangle is (AA" B" C"=AABC) by proving that C"=C. (The othercases, A" = A and B" = B, are of course analogous.)

Given: △ABC is a triangle S2 (the two-dimensional sphere). The dual point C'∈S2 of C is described by…

Answered: let AABC be a triangle S². Let the dual…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

let △ABC be a triangle S² (the two-dimensional sphere). Let the dual point C'∈S² of C be defined by the following three conditions:

(i) d(C' , A)=π/2

(ii) d(C' , B)=π/2

(iii) d(C' , C)≤π/2

A' and B' are defined analogously. Thus we get a dual triangle △A' B' C'. More precisely, △ABC is a non-degenerate triangle, and it follows (you may assume) that △A' B' C' is too.

(a) Show that the conditions (i)–(iii) determine a unique point C' by, in terms of vectors in ℝ³

thinking. (For example, (i) is in terms of perpendicularity to interpret.)

(b) Show that d(C' , X) = π/2 for all points X on the line (in the sentence of S², so great circle) line(A, B).

triangle is (△A'' B'' C''=△ABC) by proving that C''=C. (The other

cases, A'' = A and B'' = B, are of course analogous.)

I have attached an image of the questions incase the symbols dont show properly. If able please provide some explanation with the taken steps, thank you very much.