Calculus - Chapter 10 Vectors and the Geometry of Space...

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Chapter 10 Vectors and the Geometry of Space 331 Chapter 10 Section 10.1 5. 2 a –3 b 2 a – 3 b a + b b a 7. 2 a –3 b 2 a – 3 b a + b b a 9. 2, 4 3, 1 23 , 41 5, 3 += + =+ = ab 22 , 4 2 3 , 1 2, 4 6, 2 26 , 42 4,6 −= =− + 3 3 2, 4 3 2, 3 4 6, 12 == = a 525 3 , 1 2 2 , 4 15, 5 4, 8 11, 13 ba 5 2 13 11 ( 13) 290 = + = 11. (2 )( 3 )4 +=+ + − = + ab i j ij 2( 2 ) 2 ( 3) ) ( 62 ) 54 +− − − + abij i j 33 ( 2 ) 3 6 =+ = + a ( 2 ( 2 ) (15 5 ) (2 4 ) 13 9 − + = −−+ i j ij 52 1 3 9 13 ( 9) 250 51 0 = = 13. 2 ,3 1 ,0 2 1 ,3 0 3 44 1 , 0 4 1 , 4 0 4 , 0 − =− =−− = ⋅− ⋅= b 2 , 3 1 , 0 6, 9 1, 0 5, 9 += − + + 2 , 3 8 , 1 2 4 8, 12 ( 8) 12 208 4 13 = = − + = = a a 15. ) ( 3 ) 23 ( 3 ) 1 2 4 −=+ − − = −+ − −= i j bi ji j ( 2 ) ( 3 ) ( 36 3 ) 65 + + − =++− a j i j i j ( 2 ) 4 8 8 4 88 0 4 5 =+= + = ai j j
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Chapter 10 Vectors and the Geometry of Space 332 17. 0 y x a b a + b 0 y x a b a + b 19. Notice that –2 a = b . a and b are parallel. 21. If , then 2, 3 4, 6 . That is, 11 2 4 and 3 6 . This is 22 a contradiction, so and are not parallel. cc =− = §· § · −= = = ¨¸ ¨ ¸ ©¹ © ¹ ab 23. Notice that 3 a = b . a and b are parallel. 25. , then –5 = (10 2 ). That is, 1 51 0 a n d 0 2( 0 ) . T h i s 2 is a contradiction, so and are not parallel. − = = i ij 27. 52 , 43 3 , 1 AB )))& 29. 14 , 03 3 , 3 AB − =−− 31. 1(1 ) ,12 2 AB − −− = 33. 02 ,30 2 ,3 AB =−−−= −− 35. a. 4, 3 4 ( 3) 25 5 3 4, 3 , 55 5 + = = u b. 4, 3 5 , 37. a. 2 4 2 ( 4) 20 2 5 2 (2 4 ) 25 5 5 + = = ui j i j b. 12 24 2 5 , 39. a. 2 44 4 1 (4 ) 4 == i i b. 1 , 0 = i 41. a. , 21 3 , 1 31 1 0 1 31 3, 1 , 10 10 10 =+ = v v u b. 3 1 0 , 10 10 += 43. a. 2 5, 3 ( 1) 3, 4 ( 3) 4 25 5 13 4 , 5 =− − =− + = = = v v u b. 3 4 3 ,4 5 , 45. 34 2 39 1 2 555 = = + v wvi j 47. 2 9 29 29 2 29, 5 29 29 = = v wv v 49. 3 4 4, 0 3 = v 51. 0, 150 ; 20, 140 20, 10 = gr Net force is 10 pounds down, 20 pounds right. 53. 0, 200 ; 30, 10 30, 10 0, 200 30, 190 + = = gg r r Air resistance is 190 pounds up, 30 pounds right. 55. Net force 164, 115 177, 177 0, 275 13, 17 + + = Crate will move right and up. 57. , ;3 0 , 2 0 xy vw ,0w h e r e 0 ( 20) 0, so 20 300, so 300 yy < +− = = = v
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Chapter 10 Vectors and the Geometry of Space 333 2 2 400 90,000 89,600 89,600 80 14 x x x += = =− (we take the negative square root to have the plane moving westward) 80 14, 20 v This points left and up, or northwest, at an angle of 1 20 tan 3.8 80 14 ≈° north of west. 59. ,; 2 0 , 3 0 xy == vw 22 0, where 0 ( 20) 0, so 20 400, so 400 cc xx > +− = = =+ = v 2 2 400 160,000 159,600 159,600 20 399 y y y = (we take the positive square root to have the plane moving northward) 20, 20 399 = v This points right and up, or northeast, at an angle of 1 20 tan 2.9 20 399 east of north. 61. The paper travels at velocity 50, 10 .Ifthe porch is at (0, 0), then the paperboy throws from (50, c ). To travel 50 feet in the left direction, it takes 1 second. Thus in the street direction, the paper travels 10 feet. Hence the paperboy should be 10 ft up the road. 63. The force of the water is 200, 0 (magnitude of 200 lb in the direction ) 1, 0 . The weight is 0, 20 . Thus the force required to hold the hose horizontal is 200, 20 , or 200 20 20 101 pounds at 1 20 tan 5.7 200 to the horizontal. 65. The largest possible magnitude occurs when a and b are in the same direction. Thus the magnitude of a + b is the sum of the magnitudes or 7.
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This homework help was uploaded on 04/19/2008 for the course MATH 105 taught by Professor Schroder during the Spring '08 term at Millersville.

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Calculus - Chapter 10 Vectors and the Geometry of Space...

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