31_hwc1SolnsODDA

31_hwc1SolnsODDA - Therefore, the angle n between these...

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Therefore, the angle α n between these vectors is α n = 1 - ( rs ) n 1 - rs ± 1 - r 2 n 1 - r 2 ² - 1 / 2 ± 1 - s 2 n 1 - s 2 ² - 1 / 2 = (1 - r 2 ) 1 / 2 (1 - s 2 ) 1 / 2 1 - rs ± 1 - ( rs ) n (1 - r 2 n ) 1 / 2 (1 - s 2 n ) 1 / 2 ² . (c) Since both | r | < 1 and | s | < 1, we have that lim n →∞ r 2 n = 0, lim n →∞ s 2 n = 0 and also, lim n →∞ ( rs ) n = 0. Therefore lim n →∞ ` n = 1 1 - r 2 and lim n →∞ α n = 1 - r 2 1 - s 2 1 - rs . You just computed the angle between two infinite dimensional vectors! This will turn out to be more than a mere curiosity. 4.7 Consider the vectors a = h a b i . Find all vectors h c d i that are orthogonal to a . That is find conditions on c and d in terms of a and b that are necessary and sufficient for this orthogonality. SOLUTION Computing the dot product, ³ a b ´ · ³ c d ´ = ac + bd . Thus, the vectors are orthogonal if and only if ac =
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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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