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Problem 1. problem. [8 points] Given the following vectors u and v, compute the things listed in each part of this 2 u= 3 1 1 v = −1 2 (a) The dot product u · v. Solution. u · v = 2 · 1 + 3 · (−1) + 1 · 2 = 1. (b) The distance between u and v. Solution. The distance is ku − vk; since 1 u−v = 4 −1 we hare ku − vk = √ 18. (c) A unit vector in the same direction as u. Solution. This is given by 1 kuk u; we can compute kuk = √ 14 and thus the vector we want is √ 2/√14 3/ 14 . √ 1/ 14 Problem 2. [12 points] Let W be the subspace determined by the vectors w1 , w2 below (which are orthogonal to each other). Decompose the vector y given as a sum yb + z for a vector yb in W and a vector z perpendicular to W . 1 1 1 2 −1 1 y= w2 = w1 = 4 −1 1 1 1 1 Solution. The formula for yb is yb = The vector z is then y · w1 w1 · w1 w1 + y · w2 w2 · w2 1 8 1 + −4 w2 = 4 1 4 1 0 −1 z = y − yb = 1 . 0 1 1 1 −1 = 3 −1 3 1 1 .