30_hwc1SolnsODDA

30_hwc1SolnsODDA - a , and any integer n . 1 + a + a 2 +...

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v 1 · v 3 = v 3 · v 1 = 0 v 2 · v 3 = v 3 · v 2 = - 3 v 1 · v 1 = v 1 · v 1 = v 1 · v 1 = 5 . (b) | v 1 | = | v 2 | = | v 3 | = 5 . (c) The angle between v 1 and v 2 is cos - 1 (4 / 5) 0 . 6435 radians. The angle between v 1 and v 3 is cos - 1 (0) = π 2. These vectors are orthogonal. The angle between v 2 and v 3 is cos - 1 ( - 3 / 5) 0 . 9272 radians. 4.5 Fix a number r with - 1 < r < 1. For each n , let ` n be the length of the vector 1 r r 2 . . . r n - 1 in IR n . (a) Compute ` n as a function of r . (b) Now fix any other number s with - 1 < r < 1. For each n , let α n be the angle between the vectors 1 r r 2 . . . r n - 1 and 1 s s 2 . . . s n - 1 in IR n . Compute α n as a function of r and s . (c) Compute lim n →∞ ` n and lim n →∞ α n . Think about what the existence of these limits might say about the possibility of geometric considerations in infinitely many dimensions. SOLUTION (a) The formula for geometric sums says that for any number
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Unformatted text preview: a , and any integer n . 1 + a + a 2 + ··· a n-1 = 1-a n 1-a . (An easy way to derive this is to multiply the left hand side by 1-a , and notice all the cancellation that results). Applying this with a = r 2 , we get that ` n = ± 1-r 2 n 1-r 2 ² 1 / 2 . (b) Computing the dot product, 1 r r 2 . . . r n-1 · 1 s s 2 . . . s n-1 = 1 + rs + ( rs ) 2 + ··· + ( rs ) n-1 = 1-( rs ) n 1-rs . 21 /september/ 2005; 22:09 31...
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This note was uploaded on 10/03/2010 for the course MATH 380 taught by Professor Gladue during the Fall '08 term at Roger Williams.

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