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23f08HW-Answers-I - Mathematics 23 Calculus III Fall 2008...

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Mathematics 23 Calculus III Fall 2008 Some Answers I 12.1: 8. 3 , 3 5 , 6. 12. ( x - 2) 2 + ( y + 6) 2 + ( z - 4) 2 = 25; Circle in the xy -plane of radius 3 centered at (2 , - 6 , 0), circle in the yz -plane of radius 21 centered at (0 , - 6 , 4), no intersection with xz -plane. 16. ( - 4 , 3 , - 1) , 3. 32. All points outside the sphere of radius 1 centered at (0 , 0 , 1). 36. z 0 , x 2 + y 2 + z 2 4. 12.2: 10. h- 2 , 5 i . 14. h 3 , 6 i . 18. 5 i - j , 11 i - 4 j , 17 , 3 2. 22. h- 2 / 3 , 1 / 3 , 2 / 3 i . 24. h- 6 , 2 6 , 6 i . 28. 28 . 7lb,12 . 4 . 30. N70 E. 12.3: 8. - 2, 12. 1 / 2 , 0. 18. arccos(4 / 5) 37 . 22. arccos( - 3 / 102) , arccos(20 / 102) , π - arccos( - 3 / 102) - arccos(20 / 102). 26. 0 , ± 2. 38. - 37 / 7 , h 74 / 49 , - 111 / 49 , 222 / 49 i . 12.4: 2. h 10 , - 8 , 2 i . 20. h 1 / 6 , 1 / 6 , - 2 / 6 i , h- 1 / 6 , - 1 / 6 , 2 / 6 i . 28. 265. 30. h 1 , 2 , 1 i , 6 / 2. 34. 4. 38. They lie in the same plane. 12.5: 2. r = (6+ t ) i +( - 5+3 t ) j +(2 - (2 / 3) t ) k , x = (6+ t ) , y = ( - 5+3 t ) , z = (2 - (2 / 3) t ). 10. x = 2 + t, y = 1 - t, z = t, x - 2 = y - 1 - 1 = z . 14. No 16. x = 2 + t, y = 4 - t, z = 6 + 3 t, (6 , 0 , 18). 20. Skew. 22. They intersect at (3 , 5 , 1). 26. x +2 y - 3 z = - 16. 28. x + y + z = 0. 34. x - 2 y + z = 0. 44. (1 , 0 , 2). 50. Parallel. 56. x = 1+5 t, y = 1+11 t, z = 7 t, arccos(1 / 4). 12.6: 14. Ellipsoid. 16. Elliptic paraboloid. 22. IV. 24. III. 30. Elliptic cone. 13.1: 4. h 1 , 1 / 2 , 3 i . 22. I. 24. III. 13.2: 4. h 1 , 1 / (2 t ) i . 6. e t i - e - t j . 10. h sec 2 t, sec t tan t, - 2 /t 3 i . 12. (1 / 1 - t 2 ) i - ( t/ 1 - t 2 ) j . 16. a × b + 2 t ( a × c ). 20. h 1 / 2 , - 1 / 2 , 1 / 2 i . 26. x = t, y = 1 - t, z = 1 + 2 t . 34. π j + ln 2
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