6.2 Write on paper please Questions : 1,3, 5, 7, 9

Exercise Set 6.2 In Exercises 1-2, find the cosine of the angle between the vectors with In Exercises 9-10, show that the vectors are orthogonal with respect respect to the Euclidean inner product. to the standard inner product on P2 a. u = (1, -3), v = (2, 4) 9. p = -1-x+2x2, q=2x+*2 b. u = (-1, 5,2), V = (2, 4, -9) 10. p = 2- 3x + x2, 9=4+2x-2x2 c. u = (1, 0, 1,0), v = (-3, -3, -3, -3) In Exercises 11-12, show that the matrices are orthogonal with 2. a. u = (-1,0), v = (3, 8) respect to the standard inner product on M22. b. u = (4, 1, 8), v = (1,0, -3) c. u = (2, 1, 7, -1), V = (4, 0, 0,0) 1. U = [_3 3. =[-321 In Exercises 3-4, find the cosine of the angle between the vectors with respect to the standard inner product on P2. 12. U = 8 -2. V =[_13] 3. p= -1+5x+2x2, q=2+4x-9x2 In Exercises 13-14, show that the vectors are not orthogonal with 4. P = x-x2, q=7+3x+3x2 respect to the Euclidean inner product on R2, and then find a value of k for which the vectors are orthogonal with respect to the weighted In Exercises 5-6, find the cosine of the angle between A and B with Euclidean inner product (u, v) = 2u,v, + ku202. respect to the standard inner product on M22- 13. u = (1,3), v= (2, -1) 14. u = (2, -4), v= (0, 3) 5. A=[? _ 3]. B =[ 3] 15. If the vectors u = (1, 2) and v = (2, -4) are orthogonal with respect to the weighted Euclidean inner product 6. A= [_7 3]. B = [-32] ( u, v ) = willu,+ wzuzu2 In Exercises 7-8, determine whether the vectors are orthogonal with what must be true of the weights w, and wz? respect to the Euclidean inner product. 7. a. u = (-1, 3, 2), V = (4, 2, -1) 16. Let R* have the Euclidean inner product. Find two unit vectors that are orthogonal to all three of the vectors u = (2, 1, -4, 0), b. u = (-2, -2, -2), V = (1, 1, 1) v = (-1, -1, 2, 2), and w = (3, 2, 5, 4). c. u = (a, b), v = (-b,a) 17. Do there exist scalars k and I such that the vectors 8. a. u = (U1, U2, Ug), V = (0, 0,0) P1 = 2 + kx+ 6x-, p2 = 1+ 5x+ 3x2, P3 = 1+ 2x+3x2 b. u = (-4,6, -10, 1), v = (2, 1, -2, 9) c. u = (a, b, c), v = (-c, 0, a) are mutually orthogonal with respect to the standard inner product on Pz