1873_solutions

2 contains infinitely many members of s choose two

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Unformatted text preview: R 3 is defined by the equation a b  a2b3 a3b2, a3b1 a1b3, a1b2 a2b1 . a. Prove that if a and b are points in R 3 then a  a b  b  a b  0. The desired result follows at once when we work out the left side. b. Prove that if a and b are points in R 3 and t is a real number then ta b  t a b . This result follows directly from the definition. c. Prove that if a, b and c are points in R 3 then a bc  a This result follows directly from the definition. ba c d. Prove that if a and b are points in R 3 then a b 2  a 2 b 2 a  b 2. If you don’t want to work with messy computations in this exercise, let Maple do the work for you. The working steps needed for this result are somewhat messy but they are routine. e. Prove that if a and b are points in R k then a b a b. This result follows at once from part d. f. Prove that if a and b are points in R k O and  is the angle between a and b then 108 a b  a b sin . Suppose that  is the angle between a and b. Since a  b  a a b 2  a 2 b 2 ab 2 a  b 2 a Since 0 2 2 b 2  we know that sin  2 a b 2 cos 2  cos 2   a 1 0 and so aba b cos  we have 2 b 2 sin 2 . b sin . Prove that if a, b and c are points in R 3 then g. a b ca b  c. and a b c  a  c b a  b c. If you don’t want to work with messy computations in this exercise, let Scientific Notebook do the work for you. Hint: The identity a b c  ac b ab c is easy to verify for an arbitrary choice of b and c in the event that a  1, 0, 0 and a similar argument shows that the identity holds when a  0, 1, 0 or a  0, 0, 1 . We now define G to be the set of all members x of R 3 for which the identity x b c  xc b xb c 3 holds for all points b and c of R . It is easy to verify that the sum and difference of any to members of G must belong to G and that the product of any real number with a member of G must belong to G. Exercises on Convex Sets 1. Given two distinct points a and b in R k , prove that there is exactly one point x on the line segment from a to b such that x a  x b. Solution: We are being asked to show that there is just one number t 1 t a  tb a 1 t a  tb 0, 1 such that b This equation says that tb a  1 ta b which we can express as ta and so we conclude that t  1 2 b1 ta b . 2. Prove that the intersection of any two convex subsets of R k is convex. Suppose that A and B are convex subsets of R k and that a and b belong to A B and that 0 Since A is convex we know that 1 t a  tb A and since B is convex we know that 1 t a  tb B. Therefore 1 t a  tb A B. t 1. 3. Suppose that a, b and c are points in R k and that H is the set of all points of the form ra  sb  tc where r, s and t are nonnegative numbers and r  s  t  1. Prove that the set H is convex. Interpret this fact geometrically. Suppose that r 1 , r 2 , s 1 , s 2 , t 1 and t 2 are nonnegative numbers and that r 1  s 1  t 1  r 2  s 2  t 2  1, and suppose that 0 u 1. We see that 109 1 u r1a  s1b  t1c  u r2a  s2b  t2c  1 u r 1  ur 2 a  1 u s 1  us 2...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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