Determine the value (s) for k so that the angle between: v1 = (2, k) and v2 = (3, 5) is 45 degrees. show that the cauchy-schwarz inequality holds for the vectors: : a. u = (3, 2, 1) and v = (5, 2, -2) b. u = (1, 0, 2, 5) and v = (3, 1, -4, 2) c. u = (2, 1, 3, 2) and v = (6, 3, 9, 6) given the vectors: u = (2, 1, 5, 0) and v = (1, -4, 1, 2) a. determine the angle between the vectors. b. are the vectors orthogonal? why? c. determine the distance between u and v d. determine k so that ||k v || = 1 e. determine the orthogonal projection of u onto v f. show that the cauchy-schwarz inequality holds for these vectors. prove the general pythagorean theorem: if u and v are orthogonal vectors then: ||u + v||^2 = || u ||^2 + || v ||^2 prove that if w is orthogonal to both u and v then w is orthogonal to any linear combination of u and v. determine: given that: ||u + v|| = 1, ||u - v|| = 5 prove that the euclidean inner product: = sigma^n_ = 1 a b i i satisfies the properties of an inner product. use the properties of the euclidean inner product to prove that the euclidean norm satisfies the homogeneity property.