49 17. a = (1, -4, 1), b = (0, 2, -2)a=(-1,3, 4), b = (5, 2, 1)19. a = 4i - 3j+ k, b = 2i - k=8i-j+4k, b = 4j + 2k18. a20. a =21-22 Find, correct to the nearest degree, the three angles of thetriangle with the given vertices.21. P(2.0), Q(0, 3), R(3, 4)22. A(1, 0, -1), B(3, -2, 0), C(1, 3, 3)23-24 Determine whether the given vectors are orthogonal,parallel, or neither.23. (a) a = (9, 3), b = (-2, 6)(b) a = (4, 5, -2), b =(3,-1, 5)(c) a = -8i + 12j + 4k, b = 6i - 9j - 3k(d) a = 3i - j + 3k, b = 5i + 9j - 2k24. (a) u = (-5, 4, -2),v = (3, 4, -1)(b) u = 9i - 6j + 3k,(c) u = (c, c, c), v = (c, 0, -c)v = -6i+4j - 2k25. Use vectors to decide whether the triangle with verticesP(1, -3, -2), Q(2, 0, -4), and R(6, -2, -5) is right-angled.26. Find the values of x such that the angle between the vectors(2, 1,-1), and (1, x, 0) is 45°.27. Find a unit vector that is orthogonal to both i + j and i + k.28. Find two unit vectors that make an angle of 60° with= (3,4).30. x + 2y = 7,29-30 Find the acute angle between the lines.29. 2x - y = 3,3x + y = 7=205x - y = 231-32 Find the acute angles between the curves at their points ofintersection. (The angle between two curves is the angle betweentheir tangent lines at the point of intersection.)31. y = x², y = x³32. y sin x, y = cos x, 0≤x≤ π/233-37 Find the direction cosines and direction angles of the vector.(Give the direction angles correct to the nearest degree.)33. (2, 1, 2)34. (6,3, -2)35. i - 2j - 3k36. i + j + k37. (c, c, c), where c 0SECTION 12.3 The Dot Product38. If a vector has direction angles a =third direction angle y.-39-44 Find the scalar and vector projections of b onto a.39. a = (-5, 12), b = (4, 6)40. a = (1,4), b = (2,3)41. ab =(3,-1, 1)= (4,7,-4),42. a = (-1, 4, 8), b = (12, 1, 2)43. a = 3i 3j+ k,44. a = i + 2j + 3k, b = 5i - kTT/4 and 3 = π/3, find theb = 2i + 4j - k45. Show that the vector orth, b = b - projab is orthogonal to a.(It is called an orthogonal projection of b.)81346. For the vectors in Exercise 40, find orth, b and illustrate bydrawing the vectors a, b, proja b, and orth, b.47. If a =(3, 0, -1), find a vector b such that compab = 2.48. Suppose that a and b are nonzero vectors.(a) Under what circumstances iscompab comp, a?(b) Under what circumstances is proja b = projь a?=49. Find the work done by a force F = 8i - 6j +9k that movesan object from the point (0, 10, 8) to the point (6, 12, 20) alonga straight line. The distance is measured in meters and the forcein newtons.50. A tow truck drags a stalled car along a road. The chain makesan angle of 30° with the road and the tension in the chain is1500 N. How much work is done by the truck in pulling theqed car 1 km?0251. A sled is pulled along a level path through snow by a rope.A 30-lb force acting at an angle of 40° above the horizontalmoves the sled 80 ft. Find the work done by the force.52. A boat sails south with the help of a wind blowing in the direc-tion S36°E with magnitude 400 lb. Find the work done by thewind as the boat moves 120 ft.|ax₁ + by₁ +c|√a² + b²53. Use a scalar projection to show that the distance from a pointP₁(x₁, y₁) to the line ax + by + c = 0 isUse this formula to find the distance from the point (-2, 3) tothe line 3x - 4y + 5 = 0..54. If r = (x, y, z), a = (a₁, a2, a3), and b = (b₁,b2, b3), showthat the vector equation (ra) (r - b) = 0 represents asphere, and find its center and radius.55. Find the angle between a diagonal of a cube and one of itsedges.

Name: 49 17. a = (1, -4, 1), b = (0, 2, -2)a=(-1,3, 4), b = (5, 2, 1)19. a =
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Description: 49 17. a = (1, -4, 1), b = (0, 2, -2)a=(-1,3, 4), b = (5, 2, 1)19. a = 4i - 3j+ k, b = 2i - k=8i-j+4k, b = 4j + 2k18. a20. a =21-22 Find, correct to the nearest degree, the three angles of thetriangle with the given vertices.21. P(2.0), Q(0, 3), R(3,

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49 17. a = (1, -4, 1), b = (0, 2, -2)a=(-1,3, 4), b = (5, 2, 1)19. a = 4i - 3j+ k, b = 2i - k=8i-j+4k, b = 4j + 2k18. a20. a =21-22 Find, correct to the nearest degree, the three angles of thetriangle with the given vertices.21. P(2.0), Q(0, 3), R(3, 4)22. A(1, 0, -1), B(3, -2, 0), C(1, 3, 3)23-24 Determine whether the given vectors are orthogonal,parallel, or neither.23. (a) a = (9, 3), b = (-2, 6)(b) a = (4, 5, -2), b =(3,-1, 5)(c) a = -8i + 12j + 4k, b = 6i - 9j - 3k(d) a = 3i - j + 3k, b = 5i + 9j - 2k24. (a) u = (-5, 4, -2),v = (3, 4, -1)(b) u = 9i - 6j + 3k,(c) u = (c, c, c), v = (c, 0, -c)v = -6i+4j - 2k25. Use vectors to decide whether the triangle with verticesP(1, -3, -2), Q(2, 0, -4), and R(6, -2, -5) is right-angled.26. Find the values of x such that the angle between the vectors(2, 1,-1), and (1, x, 0) is 45°.27. Find a unit vector that is orthogonal to both i + j and i + k.28. Find two unit vectors that make an angle of 60° with= (3,4).30. x + 2y = 7,29-30 Find the acute angle between the lines.29. 2x - y = 3,3x + y = 7=205x - y = 231-32 Find the acute angles between the curves at their points ofintersection. (The angle between two curves is the angle betweentheir tangent lines at the point of intersection.)31. y = x², y = x³32. y sin x, y = cos x, 0≤x≤ π/233-37 Find the direction cosines and direction angles of the vector.(Give the direction angles correct to the nearest degree.)33. (2, 1, 2)34. (6,3, -2)35. i - 2j - 3k36. i + j + k37. (c, c, c), where c> 0SECTION 12.3 The Dot Product38. If a vector has direction angles a =third direction angle y.-39-44 Find the scalar and vector projections of b onto a.39. a = (-5, 12), b = (4, 6)40. a = (1,4), b = (2,3)41. ab =(3,-1, 1)= (4,7,-4),42. a = (-1, 4, 8), b = (12, 1, 2)43. a = 3i 3j+ k,44. a = i + 2j + 3k, b = 5i - kTT/4 and 3 = π/3, find theb = 2i + 4j - k45. Show that the vector orth, b = b - projab is orthogonal to a.(It is called an orthogonal projection of b.)81346. For the vectors in Exercise 40, find orth, b and illustrate bydrawing the vectors a, b, proja b, and orth, b.47. If a =(3, 0, -1), find a vector b such that compab = 2.48. Suppose that a and b are nonzero vectors.(a) Under what circumstances iscompab comp, a?(b) Under what circumstances is proja b = projь a?=49. Find the work done by a force F = 8i - 6j +9k that movesan object from the point (0, 10, 8) to the point (6, 12, 20) alonga straight line. The distance is measured in meters and the forcein newtons.50. A tow truck drags a stalled car along a road. The chain makesan angle of 30° with the road and the tension in the chain is1500 N. How much work is done by the truck in pulling theqed car 1 km?0251. A sled is pulled along a level path through snow by a rope.A 30-lb force acting at an angle of 40° above the horizontalmoves the sled 80 ft. Find the work done by the force.52. A boat sails south with the help of a wind blowing in the direc-tion S36°E with magnitude 400 lb. Find the work done by thewind as the boat moves 120 ft.|ax₁ + by₁ +c|√a² + b²53. Use a scalar projection to show that the distance from a pointP₁(x₁, y₁) to the line ax + by + c = 0 isUse this formula to find the distance from the point (-2, 3) tothe line 3x - 4y + 5 = 0..54. If r = (x, y, z), a = (a₁, a2, a3), and b = (b₁,b2, b3), showthat the vector equation (ra) (r - b) = 0 represents asphere, and find its center and radius.55. Find the angle between a diagonal of a cube and one of itsedges.

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(b) Under what circumstances is proj, b = projьa? 49. Find the work done by a force F = 8i - 6j +9k that moves an object from the point (0, 10, 8) to the point (6, 12, 20) along a straight line. The distance is measured in meters and the force in newtons.