Math Pre-Calculus Chapter 8 Review

Answers: #1-6: see online key 7. (3, 10. (0, −2) 11. ( 17. 𝑧 = 6 (cos (−3, 7𝜋 ) 4 8. (2, 10𝜋); (−2, 11𝜋) √3 1 , 2) 2 5𝜋 ) 6 14. (6, 11𝜋 ); 4 15. 𝑧 = 5√2 (cos 3𝜋 2 + 𝑖sin 7𝜋 4 + 𝑖sin 7𝜋 ) 4 16. 𝑧 = 2 (cos 18. 𝑧1 𝑧2 = −4√3 + 4𝑖; 𝑧1 1 = 2 (cos − 40° + 𝑖 sin −40°) 𝑧2 13. (√5, tan−1 (−2)) 12. (4, 4 ) 3𝜋 ) 2 19. 𝑧1 𝑧2 = −25 − 25𝑖√3; 21. −1 𝜋 22-26. see online 9. (−2, 2√3) 𝑧1 𝑧2 2𝜋 3 = 2𝑖 20. −3888 + 3888𝑖√3 27. 〈8, 6〉 28. 〈−5, −4〉 29. 〈4, −10〉, 〈0, −3〉, 〈6, −21〉, √29, 2√17, √185 30. −2𝒊 – 2𝒋, 2𝒊, 𝒊 – 3𝒋, √2, √2, 2 31. 4𝒊 – 6𝒋, −𝒊 + 𝒋, 4𝒊 – 7𝒋, √13, √5, √34 32. 〈12, −12〉, 〈−8, 5〉, 〈2, −8〉, 6√2, √5, √65 33. 〈−20, 20√3〉, −20𝒊 + 20√3j 34. 〈 35. 〈− 5√2 5√2 ,− 〉, 2 2 38. √2, 𝜋 4 − 5√2 𝐢 2 − 5√2 𝐣 2 39. 2, 4𝜋 3 + 𝑖sin 3 √3 3 √3 ,− 〉 , 𝐢 − 𝐣 2 2 2 2 36. 〈0, 2〉, 2𝒋 40. 8, 5𝜋 6 37. 1, 5𝜋 4 41. 28 42. −41 43. −4 44. −4 45. 90° 46. 180° 47. 27° 48. 71° 49. Not orthogonal 50. Orthogonal 51. −26 52. −26 53. −4 54. 168 55. 80,400 ft/lb 56. 280 ft/lb 57. 50 ft/lb 58. 3000√2 ft/lb 2𝜋 ) 3 Pre-Calculus Chapter 8 Review Name:_______________________________________ 2𝜋 3 Plot the point that has the given polar coordinates. Label each point! 𝜋 1. 𝐴 = (3, 6 ) 3. 𝐶 = (−2, 2. 𝐵 = (4, 11𝜋 ) 6 2𝜋 ) 3 4. 𝐷 = (−3, − 𝜋 2 𝜋 3 𝜋 6 5𝜋 6 3𝜋 ) 2 𝜋 5. 𝐸 = (−1, − 4𝜋 ) 3 6. 𝐹 = (4, − 0 19𝜋 ) 6 7𝜋 6 11𝜋 6 4𝜋 3 3𝜋 2 5𝜋 3 A point 𝑷(𝒓, 𝜽) is given in polar coordinates. Give two other polar representations of the point, one with 𝒓 < 𝟎 and one with 𝒓 > 𝟎. 7. (3, 3𝜋 ) 4 8. (−2, 9𝜋) Convert the polar coordinates to rectangular coordinates (𝒙, 𝒚). 9. (−4, 5𝜋 ) 3 10. (2, − 5𝜋 ) 2 11. (−1, 7𝜋 ) 6 Convert the rectangular coordinates to polar coordinates (𝒓, 𝜽) with 𝒓 > 𝟎 and 𝟎 ≤ 𝜽 < 𝟐𝝅. 12. (√8, √8) 13. (1, −2) 14. (−3√3, 3) A complex number is given. Write the number in POLAR FORM [𝒓(𝐜𝐨𝐬 𝜽 + 𝒊 𝐬𝐢𝐧 𝜽) 𝐨𝐫 𝒓 𝐜𝐢𝐬 𝜽] with argument between 𝟎 and 𝟐𝝅. 15. 5 − 5𝑖 16. −1 + √3𝑖 Find the product 𝒛𝟏 𝒛𝟐 and the quotient 2𝜋 2𝜋 + 𝑖 sin 3 ) ; 3 𝜋 𝜋 2 (cos 6 + 𝑖 sin 6 ) 18. 𝑧1 = 4 (cos 𝑧2 = 𝒛𝟏 𝒛𝟐 17. −6𝑖 . Leave your answer in 𝒂 + 𝒃𝒊 form (exact values) if possible. 19. 𝑧1 = 5(cos 100° + 𝑖 sin 100°); 𝑧2 = 10(cos 140° + 𝑖 sin 140°) Find the indicated power using DeMoivre’s Theorem. Leave your answer in 𝒂 + 𝒃𝒊 form (exact values) if possible. 5 20. (−3 − 3√3𝑖) √2 21. ( 2 + 12 √2 𝑖) 2 Sketch the vector indicated. 22. 3𝒖 23. – 𝒖 u 24. −2𝒗 25. 𝒗 – 𝒖 v 26. 𝒖 + 𝒗 Express the vector with initial point P and terminal point Q in component form. 27. 𝑃(1, 1); 𝑄(9, 7) 28. 𝑃(−1, 3); 𝑄(−6, −1) Find −𝟐𝒖, 𝒖 + 𝒗, 𝟐𝒗 – 𝒖, |𝐮|, |𝐯|, and |𝐮 − 𝐯| for the given vectors u and v. 29. 𝐮 = 〈−2,5〉, 𝐯 = 〈2, −8〉 30. 𝐮 = 𝐢 + 𝐣, 𝐯 = 𝐢 − 𝐣 31. 𝐮 = −2𝐢 + 3𝐣, 𝐯 = 𝐢 − 2𝐣 32. 𝐮 = 〈−6,6〉, 𝐯 = 〈−2, −1〉 Find the exact horizontal and vertical components of the vector with the given length and direction. Write your answer in component form AND in terms of 𝒊 and 𝒋. 33. |𝐯| = 40, 𝜃 = 2𝜋 3 35. |𝐯| = 5, 𝜃 = 225° 34. |𝐯| = √3, 𝜃 = 300° 36. |𝐯| = 2, 𝜃 = 𝜋 2 Find the magnitude and direction of the vector. Write your answer in radians. 2 2 2 2 √ √ 37. 〈− , − 〉 38. 𝐯 = 𝐢 + 𝐣 39. 𝐮 = −𝐢 − √3𝐣 40. 〈−4√3, 4〉 Find 𝐮 ∙ 𝐯 for the given vectors 𝒖 and 𝒗. 41. 𝒖 = 5𝒊 + 𝒋, 𝒗 = 6𝒊 – 2𝒋 42. 𝐮 = 〈3, −5〉, 𝐯 = 〈−2, 7〉 43. 𝐮 = 〈0, 4〉, 𝐯 = 〈10, −1〉 44. 𝒖 = – 3𝒊 – 2𝒋, 𝒗 = 2𝒋 Find the angle between 𝒖 and 𝒗 to the nearest degree. 45. 𝐮 = 𝐢 + 𝐣, 𝐯 = 𝐢 − 𝐣 46. 𝐮 = 〈−6, 6〉, 𝐯 = 〈1, −1〉 47. 𝐮 = 〈3, 4〉, 𝐯 = 〈2, 1〉 48. 𝐮 = 𝐢 + √3𝐣, 𝐯 = −√3𝐢 + 2𝐣 Determine whether the vectors are orthogonal. 49. 𝐮 = 〈5, 3〉, 𝐯 = 〈−2, 6〉 50. 𝐮 = 𝐢 − 𝐣, 𝐯 = 𝐢 + 𝐣 Find the indicated quantity, given 𝐮 = 𝟑𝐢 − 𝟐𝐣, 𝐯 = −𝟒𝐢 + 𝐣, 𝐰 = 𝟔𝐣. 51. 𝐮 ∙ 𝐯 + 𝐮 ∙ 𝐰 52. 𝐮 ∙ (𝐯 + 𝐰) 53. (𝐮 + 𝐯) ∙ (𝐮 − 𝐯) 54. (𝐮 ∙ 𝐯)(𝐮 ∙ 𝐰) Find the work done by the force F in moving an object from P to Q. 55. 𝐅 = 400𝐢 + 50𝐣; P(−1, 1), Q(200, 1) 56. 𝐅 = −4𝐢 + 20𝐣; P(0, 10), Q(5, 25) 57. The force 𝐅 = 5𝐢 − 6𝐣 moves an object 10 feet along the 𝑥-axis in the positive direction. Find the work done if the unit of force is the pound. 58. A lawn mower is pushed a distance of 100 feet along a horizontal path by a constant force of 60 lb. The handle of the lawn mower is held at an angle of 45° from the horizontal. Find the work done.