Chapter 11: Vectors and the Geometry of...

Chapter 11: Vectors and the Geometry of Space

1. Find vectors u and v whose initial and terminal points are given. Determine whether u and v are equivalent. u: (5,4), (9,14) v: (–2,5), (2,15) A) 10 4= +u i j , 10 4= +v i j , which are equivalent. B) 4 +10=−u i j , 4 +10=−v i j , which are equivalent. C) 4 –10=u i j , 4 –10=v i j , which are equivalent. D) 4 +10=u i j , 4 +10=v i j , which are equivalent. E) 4 +10=u i j , 10 4= +v j i , which are not equivalent.

2. Find the vector v whose initial and terminal points are given below.

(5,3), (7,8) A) 2 + 5=v i j D) – 2 +8=v i j B) 2 – 9=v i j E) – 2 7= −v i j C) 2 +10=v i j

3. Find the vector v whose initial and terminal points are given below.

(2.4,3.3), (–2.15,5.15) A) 4.55 + 5.15=v i j D) – 4.55 +1.85=v i j B) – 4.55 + 4=v i j E) 4.55 10= −v i j C) – 4.55 + 6=v i j

4. Find (a) 6u (b) v-u (c) 5u+2v given the following values for u and v .

3, 4 , –4,3= =u v ( a) 6u (b) v - u (c) 5u+2v A)

B)

C) 18, 24 D)

E)

36, 24 –7,1 7, 26

18, 24 –7,1 7, 23

–7, –1 7, 26

54, 24 7, –1 7, 26

18, 24 7, –1 0, 26

Copyright © Houghton Mifflin Company. All rights reserved. 323


5. The vector v and its initial point is given. Find the terminal point. –2, –1=v , initial point ( )7, 2 A) (–5, –1) B) ( )5,1 C) ( )–5,1 D) ( )5, –1 E) ( )1,5

6. Find the magnitude of the vector given below.

–4, 2=v A) 12 or 3.464102 B) 6 or 2.44949 C) 8 or 2.828427 D) 20 or 4.4721 E) 0 or 0

7. Find the unit vector in the direction of u.

–2, –3=u The possible solutions are given to two decimal places. A) –2 –3,

13 13

B) 2 3,13 13

C) 1 1,13 13

−

D) 1 1,13 13− −

E) 1 1,13 13

324 Copyright © Houghton Mifflin Company. All rights reserved.


8. Given the vectors 2,1 , 0, 2= =u v find the following: (a) (b) (c)

A) 1 0

B) 1 1

C) 0 1

D) 1 1

E) 1 0

++

u vu v

uuu + v

5 or 2.2413 or 3.6113 or 3.615 or 2.2413 or 3.61

9. Find the component form of vector v given its magnitude and the angle it makes with

the positive x-axis. 8, 260θ= =v A) 4cos(1040 ), 4sin(1040 )= °v ° D) 4sin(260 ), 4cos(260 )= ° °v B) 8sin(260 ),8cos(260 )= °v ° E) 4cos(260 ), 4sin(260 )= ° °v C) 8cos(260 ),8sin(260 )= °v °

10. Find the component form of a vector v given the magnitude of u and u+v and the

angles that u and u+v make with the positive x-axis. 2, 80 , 5, 110θ θ= = + = =u u v A) 5cos(110 ) 2sin(80 ),5cos(110 ) 2sin(80 )= ° + ° ° +v ° B) 5cos(110 ) 2cos(80 ),5sin(110 ) 2sin(80 )= ° − ° ° −v ° C) 5cos(110 ) 2sin(80 ),5cos(110 ) 2sin(80 )= ° − ° ° −v ° D) 5cos(110 ) 2cos(80 ),5sin(110 ) 2sin(80 )= ° + ° ° +v ° E) 5cos(80 ) 2sin(110 ),5cos(80 ) 2sin(110 )= ° − ° ° −v °



11. Three forces with magnitudes 75 pounds, 110 pounds and 30 pounds act on an object at angles 8 , , and , respectively, with the positive x-axis. Find the direction and magnitude of the resultant force.

0 –40 90

(The choices below are given to two decimal places.) A) 130.44 poundsat an angle of 71.18 with the positive x-axis. B) 102.78 poundsat an angle of 18.82 with the positive x-axis. C) 102.78 poundsat an angle of 71.18 with the positive x-axis. D) 102.78 poundsat an angle of –18.82 with the positive x-axis. E) 130.44 poundsat an angle of 18.82 with the positive x-axis.

12. Find the coordinates of the point that is located 5 units behind the yz-plane, 2 units in

front of the xz-plane, 3 units below the xy-plane. A) (5,2,–3) B) (–5,2,–3) C) (–5,–2,3) D) (–5,–2,–3) E) (–5,2,3)

13. Find the distance between the points given below.

(5 ,7, –1 ), ( 8 ,10 , 2 ) A) 27 or 5.1962 to four decimal places. B) 28 or 5.2915 to four decimal places. C) 26 or 5.0990 to four decimal places. D) 29 or 5.3852 to four decimal places. E) None of the above.

14. Find the coordinates of the midpoint of the line segment joining the points given below.

(6 ,3, –2 ), ( 8 ,7 , –3 ) A) ( 7 ,5 , –2.5 ) D) ( 1 ,5 , –0.5 ) B) ( 7 ,2 , –2.5 ) E) ( 1 ,2 , –2.5 ) C) ( 1 ,5 , –2.5 )

15. Find the standard equation of the sphere with center (–3 , 6, 2 ), and radius 5 .

A) ( 2 2 2+ 3) ( + 6) ( – 2) 25x y z+ + = D) 2 2 2( + 3) ( – 6) ( – 2) 25x y z+ + = B) ( 2 2 2+ 3) ( – 6) ( – 2) 5x y z+ + = E) ( 2 2 2– 3) ( – 6) ( – 2) 5x y z+ + = C) ( 2 2 2+ 3) ( – 6) ( + 2) 25x y z+ + =



16. Find the standard equation of a sphere that has diameter with the end points given below. (–2 , 8 , 4 ), ( 8 , 0 , 6 ) A) ( 2 2 2– 3) ( – 4) ( – 5) 42x y z+ + = D) 2 2 2( – 3) ( – 4) ( – 5) 10x y z+ + = B) ( 2 2 2– 3) ( + 4) ( – 5) 42x y z+ + = E) ( 2 2 2– 3) ( + 4) ( + 5) 42x y z+ + = C) ( 2 2 2+ 3) ( – 4) ( – 5) 42x y z+ + =

17. Complete the square to write the following equation in the standard equation of a

sphere.

2 2 2 + 6 + 2 – 6 15 0x y z x y z+ + + = A) ( 2 2 2+ 3) ( –1) ( – 3) 4x y z+ + = D) 2 2 2( + 3) ( +1) ( – 3) 2x y z+ + = B) 2 2 2( + 3) ( +1) ( – 3) 4x y z+ + = E) ( 2 2 2+ 3) ( –1) ( + 3) 4x y z+ + = C) ( 2 2 2– 3) ( +1) ( – 3) 4x y z+ + =

18. Find the component form of the vector u with the given initial and terminal points.

Initial point: (6 , 2 , 10 ) Terminal point: (–3 , 10 , 6 ) A) – 9 + 8 + 4=u i j k D) – 9 + 8 – 4=u i j k B) 9 + 8 – 4=u i j k E) 9 – 8 + 4=u i j k C) 3 + 12 + 16=u i j k

19. Given the vector v and its initial point find the terminal point of the vector.

1,5, – 6=v , initial point ( )–6,0, 4 A) (5, –3, –2) B) ( )5,5, –2 C) ( )–5,5, –2 D) ( )–5, –3, –2 E) None of the above



20. Find the vector z=4v+3u-4w given that: 5, 4,1=v , –1,7, –5=u , 2, –5, –2=w A) 9,57,3=z B) –9, –57, –3=z C) 9,57, –3=z D) 9, –57, –3=z E) –9,57, –3=z

21. Find the magnitude of the vector given below.

–5, –3,5=v A) 62 or 7.8740 B) 64 or 8.0000 C) 118 or 10.8628 D) 59 or 7.6811 E) 13 or 3.6056

22. Find the magnitude of the vector v given its initial and terminal points.

Initial point: (3, –3,5) Terminal point: (8, –4,2)A) 35 or 5.9161 B) 38 or 6.1644 C) 43 or 6.5574 D) 15 or 3.8730 E) 48 or 6.9282



23. Find the unit vector in the direction of u. –2,3,3=u A) –2 3 3, ,

22 22 22

D) –44 66 66, ,22 22 22

B) 2 –3 –3, ,22 22 22

E) –1 1 1, ,

22 22 22

C) 4 9 9, ,22 22 22

24. Find (a) u (b) ( (c)⋅ v )⋅u v v ( )2⋅u v given the vectors u and v.

3, 4 , 3, –3= =u v (a) u (b) ( )⋅u v v ( )2⋅ (c) u v A) 1 –6

B) –3 –4

C) –3 D) –3 –6

E) 1 –6

⋅ v

–5,13

–9,9

–9,9 9, –6

–9,9

–9,9

25. Find the angle between the vectors for u and v given below. 3, 2 , –2, 4= =vu A) 7.07 degrees D) 7.13 degrees B) 82.87 degrees E) None of the above C) 89.56 degrees

26. Find the angle between the vectors for u and v.

u i6 5 , –5 4= + = +j v i j A) –11.31 degrees D) 101.53 degrees B) –11.53 degrees E) None of the above C) 90.23 degrees



27. Determine whether u and v are orthogonal, parallel, or neither. 1, 4 , –3, –12= =u v A) Neither parallel nor orthogonal D) Parallel, same direction B) Orthogonal E) None of the above C) Parallel, opposite direction

28. Determine whether u and v are orthogonal, parallel, or neither.

u i12 6 , 2 – 4= + =j v i j A) Parallel, same direction D) Neither parallel nor orthogonal B) Parallel, opposite direction E) None of the above C) Orthogonal

29. Determine whether u and v are orthogonal, parallel, or neither.

5, –2 , 3, –10= =vu A) Orthogonal D) Parallel, opposite direction B) Parallel, same direction E) None of the above C) Neither parallel nor orthogonal

30. Find the direction cosines of the vector u given below.

3, –5,5=u A) 3 –5cos( ) , cos( ) , cos( )

59 59 59α β γ= =

5=

B) 9 25cos( ) , cos( ) , cos( )59 59 59

α β γ= =25

=

C) –3 5 –5cos( ) , cos( ) , cos( )59 59 59

α β γ= = =

D) –6 10 –10cos( ) , cos( ) , cos( )59 59 59

α β γ= = =

E) 6 –10 10cos( ) , cos( ) , cos( )59 59 59

α β γ= = =



31. Find the direction cosines of the vector u given below. u i–3 – 5 – 2= j k A) –6 –10 –4cos( ) , cos( ) , cos( )

38 38 38α β γ= = =

B) 6 10cos( ) , cos( ) , cos( )38 38 38

α β γ= =4

=

C) 3 5cos( ) , cos( ) , cos( )38 38 38

α β γ= =2

=

D) 9 25cos( ) , cos( ) , cos( )38 38 38

α β γ= =4

=

E) –3 –5 –2cos( ) , cos( ) , cos( )38 38 38

α β γ= = =

32. Find the direction angles of the vector u given below.

5, –5, –3=u A) 49.4 , –130.6 , 113.0 D) 0.9 , 130.6 , 113.0 B) 49.4 , 130.6 , 113.0 E) 49.4 , 130.6 , – 23.0 C) 113.0 , 130.6 , 49.4

33. Find the projection of u onto v, and the vector component of u orthogonal to v.

8,7 , –3,7= =vu

Projection of u onto v Component of u orthogonal to v A)

B)

C)

D)

E)

75 175– ,29 29

539 231,29 29

75 175, –58 58

539 231– , –58 58

58 87– , –13 13

539 231, –58 58

75 175– ,58 58

539 231,58 58

75 175,58 58

539 231– ,58 58



34. Find the projection of u onto v, and the vector component of u orthogonal to v. 1,6,5 , –2,10,1= =u v

Projection of u onto v Component of u orthogonal to v A)

B)

C)

D)

E)

126 630 63– , ,31 31 31

188 258 247, – ,31 31 31

63 315 63, – , –31 31 62

94 129 247– , , –31 31 62

63 315 63– , ,31 31 62

94 129 247, – ,31 31 62

63 315 63– , – , –31 31 62

94 129 247, , –31 31 62

63 315 63, ,31 31 62

94 129 247– , – ,31 31 62

35. Work An object is pulled 9 feet across a floor, using a force of 70 pounds. The

direction of the force is 60 above the horizontal. Find the work done. A) 10.47 ft-lbs D) 600.02 ft-lbs B) 315.00 ft-lbs E) None of the above C) 545.60 ft-lbs

36. Given the vectors u and v, find ×u v and ×v v .

7,6,3 , –3,10, –7= =u v

×u v A) B) C) D) E)

×v v–72, 40, 88 0, 0, 0

72, 40, 88 1, 0, 0

–72, 40, – 88 0, 0, 0

–72, 40, 88 0, 2, 0

72, – 40, 88 0, 0, 0



37. Given the vectors u nd v, find the cross product and determine whether it is orthogonal to both u and v. 6,6,5 , –1,7,8= =u v

×u v A) 13, – 53, – 48 –480 –768 Not orthogonal

B) Not orthogonal

C) 13, – 53, 48 0 0 Orthogonal

D) 13, – 53, 48 0 0 Not othogonal

E) 13, – 53, – 48 –480 –768 Orthogonal

( )× ⋅u v u ( )× ⋅u v v

–13, – 53, 48 –156 26

38. Find the area of a parallelogram that has the given vectors as adjacent sides.

6,7, 2 , –3,10, –8= =u v A) 14104 or 118.76 to two decimal places. B) 14101 or 118.75 to two decimal places. C) 28202 or 167.93 to two decimal places. D) 28204 or 167.94 to two decimal places. E) None of the above

39. Find the triple scalar product of the vectors

4,8,5 , –3, 2,5 , –7,1, –1= = =u v w

A) B) C) D) E) None of the above –275 – 276 –554 – 277

40. Use the triple scalar product to find the volume of the parallelepiped having adjacent edges given by the vectors –4,7,5 , –2, –3,8 , 3,8,0= = =u v w

A) B) C) 194 D) E) 389 47 390 291

41. Find a set of parametric equations of the line through the point ( )2,7, 2 parallel to the vector v=(8, 3,6). A) 2 + 4 , 7 +10 , 2 – 2x s y s z s= = = D) 2 8 , 7 3 , 2 6x s y s z s= + = + = + B) 2 8 , 7 3 , 2 6x s y s z s= + = − + = + E) 2 + 4 , 7 +10 , –2 – 2x s z s y s= = = C) 2 8 , 7 3 , 2 6x s y s z s= − = + = −



42. Find a set of symmetric equations of the line through the point ( )7,7, 2 parallel to the vector v=(6, 8,3). A) 7 7

–3 2 –4x y z− − −

= =2

D) 7 76 8 3

x y z 2− + += =

B) 7 76 8 3

x y z− − −= =

2 E) 7 7

–3 2 –4x z y 2− − −

= =

C) 7 76 8 3

x y z− + −= =

2

43. Find a set of parametric equations of the line through the points ( )–1,9, 2 and

. ( )5, 24,11A) –1 2 , 9 5 , 2 3x s y s z s= + = + = + D) –1+ 6 , 9 – 3 , 2 + 5x s y s z s= = = B) –1 2 , 9 5 , 2 3x s y s z s= + = − + = + E) –1+ 6 , 9 – 3 , 2 + 5x s z s y s= = = C) –1 2 , 9 5 , 2 3x s y s z s= − = + = −

44. Find a set of symmetric equations of the line through the points ( )5,7, 2 and

. ( )14,19,14A) 5 7

5 10 –5x y z− − −

= =2

D) 5 73 4 4

x y z 2+ + −= =

B) 5 75 10 –

x z y− − −= =

25

E) 5 7

3 4 4x y z 2− − −

= =

C) 5 73 4 4

x y z− + −= =

2

45. Find the set of parametric equations of the line through the point ( )1,5, 2 and is parallel

to the line x=–2 + 4t, y=3+6 t, and z=–5+8t. A) 1 4 , 5 6 , 2 8x s y s z s= + = + = + D) 1– 2 , 5 + 3 , 2 – 5x s y s z s= = = B) 1 4 , 5 6 , 2 8x s y s z s= + = − + = + E) 1– 2 , 5 + 3 , 2 – 5x s z s y s= = = C) 1 4 , 5 6 , 2 8x s y s z s= − = + = −



46. Determine whether any of the lines given below are parallel or identical.

1

2

3

4

: 4 8 , 9 7 , 7 8: 12 8 , 16 7 , 15 8: 8 , 2 7 , 1 8: 12 8 , 16 7 , 15 8

L x t y t z tL x t y t zL x t y t z tL x t y t z t

= − = − = −= + = + = += = − = += − = − = −

t

3

2 4

3

A) 1 2 4, ,L L L are identical D) 2 ,L L are identical B) 1,L L are identical E) 1,L L are parallel C) 1,L L are parallel

47. Determine whether any of the lines given below are parallel or identical.

1

2

3

4

2 10:4 7 2

0 12 6:–8 –14 4

2 1:4 7 2

0 12 6:8 14 –4

x y zL

x y zL

x y zL

x y zL

− − −= =

− − −= =

− −= =

− −− − −

= =

4

2 4

3

A) 1 2 4, ,L L L are parallel D) 2 4,L L are identical B) 1,L L are identical E) 1,L L are parallel C) 1,L L are parallel

48. Determine whether the lines given below meet. and, if so, where.

–24 8 , 10 2 , – 6 22 2 , 18 2 , 2 2

x t y t z tx s y s z s= + = + = += + = + = +

A) The lines are parallel. B) The lines are not parallel but do not intersect. C) The lines meet at (0, . 16,0)D) The lines meet at (2, . 17, –2)E) The lines meet at (–4, . 17,0)



49. Determine whether the lines given below are parallel or where they meet.

+ 9 10 + 54 6 4

5 – 272 –1 –

x y z

x y z

−= =

− −= =

52

A) The lines are parallel. D) The lines meet at (4, . 30,5)B) The lines do not intersect. E) None of the above C) The lines meet at (3, . 28,7)

50. Find an equation of a plane passing through the point given and perpendicular to the

given vector.

Point: (1,10,5) Vector –6,6,3=v A) –6 6 3 + 65 0x y z+ + = D) –6 6 3 + 69 0x y z+ + = B) –6 6 3 – 69 0x y z− + = E) –6 6 3 + 69 0x y z+ − = C) –6 6 3 – 69 0x y z+ + =

51. Find an equation of a plane passing through the following three points.

(4, –1, –4) , (–3,3,5), (2, 4, 2)

A) –21 – 24 + 27 + 0 0x y z = D) 21 + 24 + 27 + 0 0x y z =B) 21 – 24 + 27 + 0 0x y z = E) 21 – 24 – 27 + 0 0x y z = C) 21 – 24 + 27 + 0 0x y z =

52. Find an equation of a plane passing through the points

(2,3,1) , (–1, –2,10) and perpendicular to the plane 6 9 5x y z+ + + = 0 . A) –2 + 3 0x y z+ = D) –2 + 3 6 0x y z+ − = B) –2 + 3 –6x y z+ = E) –2 + 3 2 –6x y z+ = C) –2 + 3 2 6 0x y z+ − =



53. Determine whether the following planes are parallel, orthogonal, or neither. If they are neither parallel nor orthogonal, find the angle of intersection. –5 –14 8 3 0

2 3 2 0

x y z

x y z

+ + =

+ + − =

A) The planes are parallel. B) The planes are orthogonal. C) The planes are neither parallel nor orthogonal, the angle of intersection is 105 . D) The planes are neither parallel nor orthogonal, the angle of intersection is 120 . E) None of the above

54. Determine whether the following planes are parallel, orthogonal, or neither. If they are

neither parallel nor orthogonal, find the angle of intersection. 8 9 2 5 0

–3 13 5 3 0

x y z

x y z

− + + =

+ − − =

A) The planes are parallel. B) The planes are orthogonal. C) The planes are neither parallel nor orthogonal, the angle of intersection is . 29.75D) The planes are neither parallel nor orthogonal, the angle of intersection is . 26.84E) None of the above

55. Find the distance between the point (2, 0, 0) and the plane given below.

–2 3 2 –10x y z− + =

A) 6

17 B) –6

17 C) 11

17 D) 12

17 E) –12

17

56. Find the distance between the planes given below.

2 3 3 29

4 6 6 24

x y z

x y z

− + − =

− + − =

0

0

A) –17

22 B) 34

22 C) –34

22 D) 17

22 E) 22

22



57. Identify the following quadric surface.

2 2 2

15 6 14x y z

+ + =

A) Ellipsoid D) Elliptic cone B) Hyperboloid of one sheet E) Elliptic paraboloid C) Hyperboloid of two sheets F) Hyperbolic paraboloid


2 2 2

18 2 2x y z

+ − =



2 2 2

15 10 6x y z

− − =



2 2 2

05 6 2x y z

+ − =



2 2

4 2x yz = +





2 2

7 4x yz = −


63. Find an equation of the surface of revolution generated by revolving the curve given

below in the indicated coordinate plane about the given axis. Equation of Curve Coordinate Plane Axis of Revolution

2 4=xx

4=

z x xz-plane x-axis A) 2 2 4z y+ = D) 2 4z y= B) 2 2 4z x+ = E) None of the above C) 2 4z x=

64. Find an equation of the surface of revolution generated by revolving the curve given

below in the indicated coordinate plane about the given axis. Equation of Curve Coordinate Plane Axis of Revolution z y yz-plane z-axis A) 2 2

4zx y+ =

D) 22 2

16yx z+ =

B) 22 2

4zx y+ =

E) 22 2

16xy z+ =

C) 22 2

16zx y+ =



65. Find an equation of a generating curve given the equation of its surface of revolution is

22 2

9zx y+ =

Equation of Curve Coordinate Plane Axis of Revolution A) 3=z y yz-plane z-axis B) 3=y z yz-plane y-axis C) 3=

3=y z xy-plane z-axis

D) z x yz-plane y-axis E) 3x z= xz-plane z-axis

66. Find the equation of the surface satisfying the conditions, and identify the surface.

The set of points equidistant from the point (7, 3, 4 ) and the plane y=-4. A) ( ) ( )2 214 – 7 3 – 7y x z= + − , an elliptic paraboloid B) ( ) ( ) ( )2 2 2– 7 3 4 7 0x y z+ − + − − = , a sphere C) ( ) ( )2 214 – 7 3 – 4y x z= − − , a hyperbolic paraboloid D) ( ) ( ) ( )2 2 2– 7 3 4 4 0x y z+ − + − − = , a sphere E) None of the above

67. Convert the following point from cylindrical coordinates to rectangular coordinates.

58, , 33π

A) ( )16, –16 3,3 D) ( )4, –4 3,3

B) ( )–8 3,8,3 E) ( )8, –8 3,3

C) ( )–4 3,4,3



68. Convert the following point from rectangular coordinates to cylindrical coordinates. Give any angles in radians. ( )8, –8,8 A) 78 2, , 8

4π

D) 716 2, , 8

4π

B) 78, , 84π

E) 78 2, , 8

2π

C) 78, , 82π

69. Find an equation in cylindrical coordinates for the equation given in rectangular

coordinates.

2 29 9z x y= + −8 A) 29 8

8z r= +

D) 9 8z r= +

B) 29 88

z r= − E) None of the above

C) 29 8z r= +

70. Find an equation in cylindrical coordinates for the equation given in rectangular coordinates.

2 29 9 4x y x+ = A) 9 cos

4r θ=

D) 14 cos9

r θ−=

B) 4 cos9

r θ= E) None of the above

C) 4 sin9

r θ=



71. Find an equation in cylindrical coordinates for the equation given in rectangular coordinates.

2 2 249 49 2 Tx y z+ − = A) 2 49 T

2rz −

= D) 2

2 T 492

rz −=

B) 249 T2

rz −=

E) 22

249 T

zr

=−

C) 22 49 T

2rz −

=

72. Find an equation in rectangular coordinates for the equation given in cylindrical

coordinates.

6sinr θ= A) 2 2 6x y x+ = D) 2 2

6xx y+ =

B) 2 2 6x y y+ = E) 2 2

6yx y+ =

C) 2 2 6x z x+ =

73. Find an equation in rectangular coordinates for the equation given in cylindrical coordinates

6r z= A) 2 2 6x y z+ = D) 2 2

6yx y+ =

B) 2 2 36 2x y z+ = E) 2 2

6zx y+ =

C) 2 2 6x z y+ =

74. Find an equation in rectangular coordinates for the equation given in cylindrical coordinates.

2 2 49r z+ = A) 7x y z+ + = D) 2 2 2 49x y z+ + = B) 2 2 49 2x y z+ = − E) 2

2 2 07zx y + + =

C) 2 2 2 7x y z+ + =



75. Convert the point from spherical coordinates to rectangular coordinates.

23, ,3 6π π

A) 3 3 3 3 3, , –4 4 2

D) 3 3 3 3 3, – ,

4 4 2

B) 3 3 3 3 3– , – , –4 4 2

E) 3 3 3 3 3, – , –

4 4 2

C) 3 3 3 3 3– , ,4 4 2

76. Find an equation in spherical coordinates for the equation given in rectangular

coordinates.

6y = A) 6sec cscρ φ θ= D) 6sec secρ φ θ= B) 6sin sinρ φ θ= E) 6cos cosρ φ θ= C) 6csc cscρ φ θ=

77. Find an equation in spherical coordinates for the equation given in rectangular

coordinates.

2 2 2+ 4 7x y z+ = A)

( )2

2 2

7sin + 4cos

ρφ φ

= D)

( )2

2 2

7cos + 4sin

ρφ φ

=

B) 7+4

ρ = E)

( )2

2 2

7sin + 4cos

ρφ θ

=

C) ( )

22 2

7sin + 4cos

ρθ θ

=

78. Find an equation in rectangular coordinates for the equation given in spherical

coordinates.

π8

θ =

The coefficients below are given to two decimal places.

A) B) 0.41y = z 0.41x y= C) 0.41y x= D) 2.41y x= E) 0.41z x=



79. Find an equation in rectangular coordinates for the equation given in spherical coordinates.

6csc cscρ ϕ θ= A) 1

6y = B) 1

6x = C) 6x = D) 6y = E) 6z =

80. Convert the following point from cylindrical coordinates to spherical coordinates.

π4 3, , 44

A) π 28, ,4 3

π

D) π 24, ,

4 3π

B) π8, ,4 3π

E) π4 3, ,

4 3π

C) π4, ,4 3π

81. Convert the following point from spherical coordinates to cylindrical coordinates.

π 5π9, ,8 6

A) π –9 39, ,8 2

D) 9 π 9 3, ,

2 8 2

B) 9 3 π 9, ,2 8 2

E) 9 π –9 3, ,

2 8 2

C) –9 3 π 9, ,2 8 2




82. Identify the solid that has the given description in spherical coordinates.

π π π1, 0 ,2 4 2

ρ θ φ= ≤ ≤ ≤ ≤

A)

B)

C)

D)

E)


Answer Key

1. D Section: 11.1

2. A Section: 11.1

3. D Section: 11.1

4. C Section: 11.1

5. B Section: 11.1

6. D Section: 11.1

7. A Section: 11.1

8. B Section: 11.1

9. C Section: 11.1

10. B Section: 11.1

11. B Section: 11.1

12. B Section: 11.2

13. A Section: 11.2

14. A Section: 11.2

15. D Section: 11.2

16. A Section: 11.2

17. B Section: 11.2

18. D Section: 11.2

19. C Section: 11.2

20. C Section: 11.2

21. D Section: 11.2

22. A Section: 11.2



23. A Section: 11.2

24. D Section: 11.3

25. B Section: 11.3

26. D Section: 11.3

27. C Section: 11.3

28. C Section: 11.3

29. C Section: 11.3

30. A Section: 11.3

31. E Section: 11.3

32. B Section: 11.3

33. D Section: 11.3

34. C Section: 11.3

35. B Section: 11.3

36. A Section: 11.4

37. C Section: 11.4

38. B Section: 11.4

39. D Section: 11.4

40. A Section: 11.4

41. D Section: 11.5

42. B Section: 11.5

43. A Section: 11.5

44. E Section: 11.5

45. A Section: 11.5



46. A Section: 11.5

47. D Section: 11.5

48. C Section: 11.5

49. C Section: 11.5

50. C Section: 11.5

51. B Section: 11.5

52. D Section: 11.5

53. B Section: 11.5

54. C Section: 11.5

55. A Section: 11.5

56. D Section: 11.5

57. A Section: 11.6

58. B Section: 11.6

59. C Section: 11.6

60. D Section: 11.6

61. E Section: 11.6

62. F Section: 11.6

63. A Section: 11.6

64. C Section: 11.6

65. A Section: 11.6

66. A Section: 11.6

67. D Section: 11.7

68. A Section: 11.7




69. E Section: 11.7

70. B Section: 11.7

71. C Section: 11.7

72. B Section: 11.7

73. B Section: 11.7

74. D Section: 11.7

75. C Section: 11.7

76. C Section: 11.7

77. A Section: 11.7

78. C Section: 11.7

79. D Section: 11.7

80. B Section: 11.7

81. E Section: 11.7

82. C Section: 11.7

Chapter 11: Vectors and the Geometry of...

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