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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 - 1 / 2 1 - s 2 n 1 - s 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 . (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 = a b . Find all vectors c d 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, a b · c d = ac + bd . Thus, the vectors are orthogonal if and only if
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