Exam - Okan Üniversitesiusers.okan.edu.tr/sezgin.sezer/public_html/Test 3.pdf · 2015-11-24 · 11)z = x2 + 4y2 11) Determine whether the following is always true or not always true

Exam

Name___________________________________

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Solve the problem.

1) Show that the midpoint of the hypotenuse of a right triangle is equidistant from all three

vertices. [Hint: See the figure below. Show that a + b2

= a - b2

.]

1)

Determine whether the following is always true or not always true. Given reasons for your answers.

2) (u ˛ v) œ v = u œ (u ˛ v) 2)

Sketch the given surface.

3) x2 + y2 = 4 3)


4) u ˛ (v + w) = u ˛ v + u ˛ w 4)

Solve the problem.

5) Show that A = au + bv is orthogonal to B = bu - av, where u and v are orthogonal unitvectors.

5)


6) (u ˛ v) œ w = u œ (w ˛ v) 6)

1


7) x2 + y2 = z2 7)

Solve the problem.

8) Shade in the points (x, y) for which (xi + yj) œ v ≥ 0. Justify your answer. 8)


9) (u ˛ v) œ v = 0 9)


10) x = 1 - y2 - z2 10)

2

11) z = x2 + 4y2 11)


12) c(u ˛ v) = cu ˛ cv (any number c) 12)

Sketch the coordinate axes and then include the vectors A, B, and A ˛̨̨̨ B as vectors starting at the origin.

13) u = 2i + j, v = i - 2j 13)

14) u = i - j, v = k 14)


15) y = x2 15)

3

16) 16x2 + y2 + z2 = 16 16)


17) u ˛ v = -(v ˛ u) 17)

Solve the problem.

18) Show that the vectors a b + b a and a b - b a are orthogonal. 18)


19) u = i + k, v = i - k 19)


20) |u| = u œu 20)

21) c(u œ v) = cu œ cv (any number c) 21)

Solve the problem.

22) The unit vectors u and v are combined to produce two new vectors a = u + v and b = u - v.Show that a and b are orthogonal. Assume u ≠ v.

22)


23) u ˛ 0 = 0 23)


24) u = i, v = k 24)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the triple scalar product (u x v) œœœœ w of the given vectors.

25) u = 2i - 4j + 3j; v = -4i - 6j + 4k; w = 9i - 7j + 3k

A) 140 B) -74 C) 458 D) 74

25)

4

Find the vector projv u.

26) v = 7i - 3j + k, u = -4j + 3k

A) - 6059

j + 4559

k B) 215

i - 95j + 35k

C) - 125

j + 95k D) 105

59i - 4559

j + 1559

k

26)

Find v œœœœ u.

27) v = 13, 111 and u = 1

3, -111

A) 13 - 111

B) 0 C) 13i - 111

j D) 23i

27)

Find the indicated vector.

28) Let u = 5, 7 , v = -4, -1 . Find v - u.

A) 2, 3 B) -9, -8 C) 1, 6 D) -6, -11

28)

Identify the type of surface represented by the given equation.

29) x22 + y

29 + z

27 = 1

A) Elliptic cone

B) Sphere

C) Ellipsoid

D) Paraboloid

29)

Solve the problem.

30) Find the magnitude of the torque in foot-pounds at point P for the following lever:

PQ = 4 in. and F = 15 lb

A) -950.70 ft-lb B) 3407.50 ft-lb C) -3002.23 ft-lb D) 60 ft-lb

30)

Find the center and radius of the sphere.

31) 3x2 + 3y2 + 3z2 - 2x + 2y = 9

A) C 13, - 13, 0 , a = 29

3B) C - 1

9, 19, 0 , a = 29

9

C) C - 13, 13, 0 , a = 29

9D) C - 1

3, 13, 0 , a = 29

3

31)

5

Describe the given set of points with a single equation or with a pair of equations.

32) The plane perpendicular to the y-axis and passing through the point (4, -2, -4)

A) y = -2 B) 4x - 4z = 0 and y = -2

C) x - 2 + z = 0 D) x = 4 and z = -4

32)

Write u as the sum of a vector parallel to v and a vector orthogonal to v.

33) u = -8i + 9k, v = 4i + 3j

A) u = - 12825

i - 9625

j + - 7225

i + 9625

j + 9k B) u = 25625

i - 28825

k + 35625

i + 3j - 12825

k

C) u = - 12825

i - 9625

j + - 32825

i - 9625

j + 9k D) u = 25625

i - 28825

k + - 15625

i + 3j + 12825

k

33)

Find the length and direction (when defined) of u ˛̨̨̨ v.

34) u = -5i - 2j - 2k, v = 10i + 4j + 4k

A) 0; 0 B) 165; 1165

(5i + 2j + 2k)

C) 3 11; 13 11

(5i + 2j + 2k) D) 0; 5i+ 2j + 2k

34)

Find an equation for the sphere with the given center and radius.

35) Center (-10, 0, 0), radius = 8

A) x2 + y2 + z2 + 20x = -36 B) x2 + y2 + z2 - 20x = 8

C) x2 + y2 + z2 - 20x = -36 D) x2 + y2 + z2 + 20x = 8

35)


36) u = - 12i + 32j + k, v = i + j + 2k

A) 8; 14i - 14j + 12k B) 2 3; 3

3i + 3

3j - 3

3k

C) 2 2; - 22

i + 22

j - 22

k D) 8; 12i - 14j - 14k

36)

Solve the problem.

37) Find the volume of the solid bounded by the ellipsoid x216 + y

281 + z

225 = 1 and the planes z = -2 and

z = 2. (The area of an ellipse with semiaxes a and b is pab)

A) 105.60p units3 B) 4907.52p units3 C) 136.32p units3 D) 151.68p units3

37)


38) x2 + y2 + z2 - 16x - 8y + 16z = -95

A) C(-8, -4, 8), a = 7 B) C(8, 4, -8), a = 49

C) C(8, 4, -8), a = 7 D) C(8, 4, 8), a = 7

38)

6


39) x27 - y

28 - z

23 = 1

A) Elliptic cone B) Hyperboloid of one sheet

C) Hyperboloid of two sheets D) Ellipsoid

39)

Find the component form of the specified vector.

40) The unit vector that makes an angle -5p/3 with the positive x-axis

A) 12, 12

B) 12, 32

C) -12, - 32

D) - 32, -12

40)

Find parametric equations for the line described below.

41) The line through the point P(-2, -5, -2) and parallel to the line x = 4t - 2, y = 3t + 6, z = 2t - 5

A) x = 4t + 2, y =3t + 5, z = 2t + 2 B) x = 3t - 2, y = -4t, z = -2

C) x = 4t - 2, y = 3t - 5, z = 2t - 2 D) x = -2, y = 2t - 5, z = -3t - 2

41)


42) u = i + j + j; v = 3i + 8j + 4k; w = 10i + 6j + 3k

A) 97 B) -31 C) 93 D) -93

42)


43) The circle of radius 5 centered at the point (0, -10, -5) and lying in a plane parallel to the xy plane

A) (x - 0)2 + (y - 10)2 = 52 and x + y = -10 B) x02+ y-10

2= 52 and z = -5

C) (x - 0)2 + (y - 10)2 = 52 and z = -5 D) x02+ y-10

2+ 1 = 52

43)

Calculate the requested distance.

44) The distance from the point S(7, 1, -5) to the line x = -6 + 3t, y = -9 + 12t, z = 10 + 4t

A) 73685169

B) 73,685169

C) 7368513

D) 73,68513

44)


45) The vector PQ , where P = (-8, -2) and Q = (-4, 3)

A) -10, 3 B) -12, 1 C) 4, 5 D) -4, -5

45)

Find v œœœœ u.

46) v = 4i - 2j and u = -7i - 8j

A) -28i + 16j B) -3i - 10j C) -44 D) -12

46)

Find the acute angle between the lines.

47) -3x - 5y = 9 and 8x - 6y = -8

A) 1.468 radians B) 0.1031 radians C) 3.039 radians D) 0.1029 radians

47)

Find v œœœœ u.

48) v = 3i - 8j and u = -4 13i + 2j

A) -16 13i - 12j B) (3 - 4 13)i - 6j C) -16 - 12 13 D) -16 13 + 12

48)

7


49) The distance from the point S(5, 2, 3) to the plane 2x + 2y + z = 7

A) 6 B) 2 C) 109

D) 103

49)

Match the equation with the surface it defines.

50) x2

32 + z

2

32 = y3

A) Figure 3 B) Figure 2 C) Figure 1 D) Figure 4

50)

Express the vector in the form v = v = v1i + v2j + v3k.

51) AB if A is the point (-6, -4, 4) and B is the point (-1, -11, 7)

A) v = 5i - 7j + 3k B) v = 5i + 7j - 3k C) v = 5i + 7j + 3k D) v = 5i - 7j - 3k

51)

Write the equation for the plane.

52) The plane through the point A(9, 7, 4) perpendicular to the vector from the origin to A.

A) 9x + 7y + 4z = 146 B) 9x + 7y + 4z = -146

C) 9x + 7y + 4z = 146 D) 9x + 7y + 4z = 20

52)


53) v = k, u = 6i + 9j + 2k

A) 2121

k B) 12121

i + 18121

j + 4121

k

C) 1211

i + 1811

j + 411

k D) 2k

53)


54) The plane through the points P(5, -7, -20) , Q(-3, 6, -6) and R(-1, -4, 4).

A) 2x + y + 5z = -9 B) 5x + 2y + z = 9 C) 5x + 2y + z = -9 D) 2x + y + 5z = 9

54)

8

Find an equation for the line that passes through the given point and satisfies the given conditions.

55) P = (12, 12); perpendicular to v = 5i - 6j

A) 5x - 6y = 61 B) y - 12 = 187(x - 5)

C) 5x - 6y = -12 D) -6x - 5y = -132

55)


56) u = 4i + 2j + 8k, v = -i - 2j - 2k

A) 180; 2 515

i + 1515

j + 515

k B) 6 5; 2 55

i - 55

k

C) 180; 115

i + 130

k D) 6 5; 2 55

i + 55

k

56)

Give a geometric description of the set of points whose coordinates satisfy the given conditions.

57) x2 + y2 ≤ 16, z = -10

A) All points on or outside of the circle x2 + y2 = 16 and in the plane z = -10

B) All points on the cylinder with radius 4 along the z-axis

C) All points on or within the circle x2 + y2 = 16 and in the plane z = -10

D) All points within the parabola x2 + y2 = 16 in the plane z = -10

57)


58) 3x - y = 16 and 2x + y = -5

A) 45e B) 75e C) 60e D) 30e

58)

Express the vector as a product of its length and direction.

59) 1i + 2j + 2k

A) 3 13i + 23j + 23k B) 3(i + j + k) C) 3 1

9i + 29j + 29k D) 3(1i + 2j + 2k)

59)


60) P = (-8, -2); parallel to v = -2i + 6j

A) -2x + 6y = 4 B) -2x + 6y = 40 C) 6x + 2y = -52 D) y + 2 = 43(x + 2)

60)


61) u = 2i + 2j - k, v = -i + k

A) 3; - 23i + j - 2

3k B) 9; 2

9i - j + 2

9k C) 9; 2

9i - j - 2

9k D) 3; 2

3i - j + 2

3k

61)

Find the angle between the planes.

62) 5x - 4y + 5z = 1 and -3x - 9y + 3z = -9

A) 0.616 B) 0.462 C) 1.109 D) 1.497

62)

Find the intersection.

63) x + y + z = -1, x + y = 5

A) x = t, y = 5 - t, z = -6 B) x = -t, y = 5 + t, z = -6

C) x = -t, y = 5 + t, z = 6 D) x = -1, y = 1 + 5t, z = -6t

63)

9


64) x = -5z2, no limit on y

A) Parabolic cylinder B) Cylinder

C) Hyperboloid of two sheets D) Sphere

64)

Write one or more inequalities that describe the set of points.

65) The seventh octant of the xyz coordinate system

A) x ≥ 0, y ≥ 0, z ≤ 0 B) x ≤ 0, y ≥ 0, z ≥ 0 C) x ≥ 0, y ≥ 0, z ≥ 0 D) x ≤ 0, y ≤ 0, z ≤ 0

65)


66) Let u = -8, -5 , v = -2, 5 . Find 45u + 3

5v.

A) - 1, - 385

B) - 385, - 1 C) - 8, 0 D) - 47

5, 75

66)


67) x = -2 - 2t, y = -5 + 2t, z = -3 + 2t ; 1x - 3y - 7z = 7

A) 511, - 8211, - 6011

B) - 4911, - 2811, - 611

C) (0, -7, -5) D) (-4, -3, -1)

67)


68) 2x - 6y = -5 and 2x - 2y = -1


68)

69) 2x + 3y = -8 and 3x + 7y = 6


69)


70) -4i + 4j + 4k

A) 4 3 - 33

i + 33

j + 33

k B) 4 3 - 148

i + 148

j + 148

k

C) 4( -i + j + k) D) 312

- 33

i + 33

j + 33

k

70)


71) The line through the point P(-7, -2, -2) and perpendicular to the vectors u = -8i + 3j - 5k andv = 5i + 3j - 3k

A) x = 16t - 7, y = -49t - 2, z = 2t - 2 B) x = 16t - 7, y = 49t - 2, z = -39t - 2

C) x = 16t - 7, y = -49t - 2, z = -39t - 2 D) x = 16t + 7, y = -49t + 2, z = 2t + 2

71)


72) The circle of radius 4 centered at the point (9, -2, 1) and lying in a plane perpendicular to the x-axis

A) (y + 2)2 + (z - 1)2 = 42 and y + z = -1 B) y-22+ z12+ 1 = 42

C) y-22+ z12 = 42 and x = 9 D) (y + 2)2 + (z - 1)2 = 42 and x = 9

72)

10


73) Let u = 4, -8 . Find 8u.

A) 32, -64 B) -32, 64 C) -32, -64 D) 32, 64

73)

Find the distance between points P1 and P2.

74) P1(-1, -2, 6) and P2(9, 3, 16)

A) 14 B) 15 C) 10 D) 18

74)


75) The plane through the point (6, -6, -5) and perpendicular to the x-axis

A) x = 6 B) y + z = -11 C) y = -6 and z = -5 D) 6 + y + z = 0

75)

Solve the problem.

76) An airplane is flying in the direction 37e east of south at 626 km/hr. Find the component form ofthe velocity of the airplane, assuming that the positive x-axis represents due east and the positivey-axis represents due north.

A) 499.9, -376.7 B) 0.6018, -0.7986 C) -402.9, -479.1 D) 376.7, -499.9

76)


77) P1(-4, 6, 5) and P2(-1, 1, 1)

A) 25 B) 5 2; C) 10 D) 50

77)

Solve the problem.

78) Find a formula for the distance from the point P(x, y, z) to the xy plane.

A) z B) x2 + y2 C) y D) x

78)


79) 87j + 15

7k

A) 17787j + 15

7k B) 17

7(j + k) C) 289

4987j + 15

7k D) 17

7817

j + 1517

k

79)


80) x29 + z

210 = y4

A) Elliptic cone B) Ellipsoid

C) Elliptic paraboloid D) Hyperbolic paraboloid

80)

Find a parametrization for the line segment beginning at P1 and ending at P2.

81) P1(1, -2, 2) and P2 0, -2, 14

A) x = -1t + 1, y = -2t, z = - 74t + 2 B) x = -1t + 1, y = -2, z = - 7

4t + 2

C) x = 1t, y = -2t, z = 74t + 1 D) x = 1t, y = -2, z = 7

4t + 1

81)

11


82) The line through the points P(-1, -1, 4) and Q(3, 1, 1)

A) x = t - 4, y = t - 2, z = 4t + 3

B) x = 4t - 1, y = 2t - 1, z = -3t + 4

C) x = 4t + 1, y = 2t + 1, z = -3t - 4

82)


83) v = 2i - 2j - 4k, u = 5i - 12k

A) 2912

i - 2912

j - 2912

k B) 296

i - 296

j - 293

k C) 29013

i - 696169

k D) 290169

i - 696169

k

83)


84) - 32i - 6j + 2k

A) 132(-i - j + k) B) 169

4- 313

i - 1213

j + 413

k

C) 213

D) 132- 313

i - 1213

j + 413

k

84)


85) The plane through the point P(3, 2, -2) and parallel to the plane 4x + 5y + 3z = 17.

A) 4x + 5y + 3z = -16 B) 5x + 3y + 4z = 16

C) 3x + 2y - 2z = 16 D) 4x + 5y + 3z = 16

85)

Find the angle between u and v in radians.

86) u = 4i, v = 8i - 10j

A) 1.39 B) 0.67 C) 0.90 D) 1.56

86)


87) The plane through the point P(-10, -7, 9) and perpendicular to the line x = 7 + 7t, y = -5 + 9t,z = 7 - t.

A) 7x + 9y - z = 25 B) 7x + 9y + z = -142

C) 7x + 9y - z = -142 D) 7x + 9y - z = 142

87)


88) P = (9, 6); perpendicular to v = 4i + 2j

A) 4x + 2y = 48 B) y - 6 = 45(x - 4) C) 2x - 4y = -6 D) 4x + 2y = 20

88)


89) - 4i - 53j

A) 133( -i - j) B) 13

3 - 1213

i - 513

j C) 1699- 4i - 5

3j D) 13

3- 4i - 5

3j

89)

12


90) v = i + j + k, u = 3i + 4j + 12k

A) 19169

i + 19169

j + 19169

k B) 193

i + 193

j + 193

k

C) 1913

i + 1913

j + 1913

k D) 203

i + 203

j + 203

k

90)


91) 5x - 2y = 8, -4y + 8z = -7

A) x = -16t - 46, y = -40t - 7, z = -20t B) x = -16t - 2310, y = -40t - 7

4, z = 20t

C) x = -16t + 2310, y = -40t + 7

4, z = -20 D) x = -16t + 23

10, y = -40t + 7

4, z = -20t

91)

Solve the problem.

92) Find the area of the triangle determined by the points P(1, 1, 1), Q(-9, 5, 9), and R(-10, 1, 5).

A) 46822

B) 27,5822

C) 27,582 D) 4682

92)

93) A bird flies from its nest 7 km in the direction 34e north of east, where it stops to rest on a tree. Itthen flies 10 km in the direction 22e south of west and lands atop a telephone pole. With anxy-coordinate system where the origin is the bird's nest, the x-axis points east, and the y-axispoints north, at what point is the tree located?

A) (-5.940, 3.704) B) (3.914, 5.803) C) (5.803, 3.914) D) (0.8290, 0.5592)

93)


94) y2

82 + z

2

42 = 1


94)

13


95) P1(0, 0, 0) and P2(-4, 5, -5)

A) x = -4t, y = 5t, z = -5t B) x = -4t + 4, y = 5t - 5, z = -5t + 5

C) x = 4t, y = -5t, z = 5t D) x = -4t - 4, y = 5t + 5, z = -5t - 5

95)


96) The distance from the point S(3, 10, 7) to the line x = 7 + 2t, y = -7 + 6t, z = 10 + 9t

A) 33505121

B) 33,50511

C) 3350511

D) 33,505121

96)


97) u = 10i + 10j + 5k, v = 7i + 9j + 6k

A) 1.06 B) 0.18 C) 1.50 D) 1.39

97)

Find the angle between the curves.

98) y = 7x2 and y = 7x3


98)

Solve the problem.

99) Find the volume of the solid bounded by the hyperboloid of one sheet x29 + y

264 - z

216 = 1 and the

planes z = -4 and z = 4. (The area of an ellipse with semiaxes a and b is pab)

A) 208.00p units3 B) 128.00p units3 C) 256.00p units3 D) 6144.00p units3

99)


100) u = 4i + 2j - j; v = 7i + 5j - 2k; w = 3i + 7j - 8k

A) -156 B) -238 C) 58 D) -38

100)

Solve the problem.

101) Find the vector from the origin to the center of mass of a thin triangular plate (uniform density)whose vertices are A(3, 5, 4), B(6, 9, 10), and C(2, 10, 2).

A) 52i - 3j + 1k B) 11

3i + 8j + 16

3k C) 5

3i - 2j + 2

3k D) 5

6i - 1j + 1

3k

101)


102) The plane through the point (5, 9, 3) and parallel to the yz-plane

A) 5 + y + z = 0 B) 9y + 3z = 0 and x = 5

C) y = 9 and z = 3 D) x = 5

102)

103) The set of points equidistant from the points (-3, 0, 0) and (9, 0, 0)

A) x = 3 B) y + z = 0 and -3 < x < 9

C) x > -3 and x < 9 D) y + z = 3

103)

Solve the problem.

104) Let u = 4i + j, v = i + j, and w = i - j. Find scalars a and b such that u = av + bw.

A) 4v -1w B) 4v + 1w

C) 0.4000 v + 0.6667 w D) 2.500 v + 1.500w

104)

14


105) - x2

62 + y

2

32 + z

29 = 1


105)


106) 3x - 3y - 6z = 10and- 7x + 8y - 8z = 8

A) 1.093 B) 1.540 C) 0.031 D) 1.510

106)

Solve the problem.

107) Find the area of the parallelogram determined by the points P(7, -5, 5), Q(-7, -10, -9), R(-4, -6, -7)and S(-18, -11, -21).

A) 113,9212

B) 11 332

C) 11 33 D) 113,921

107)


108) x = -2 + 3t, y = 9 + 4t, z = 1 + 10t ; 8x + 3y + 6z = 5

A) - 198, 52, - 14

B) (1, 13, 11) C) (-5, 5, -9) D) - 138, 72, 94

108)


109) Let u = -9, 6 . Find -2u.

A) -18, 12 B) -18, -12 C) 18, 12 D) 18, -12

109)

110) Let u = -6, -4 . Find -6u.

A) 36, -24 B) -36, 24 C) 36, 24 D) -36, -24

110)


111) P = (-8, -5); perpendicular to v = -6i + 9j

A) 9x + 6y = -102 B) -6x + 9y = 117 C) y + 5 = 7(x + 6) D) -6x + 9y = 3

111)

15

Solve the problem.

112) Find the volume of the solid bounded by the elliptical cone x281 + y

264 = z

225 and the planes z = 0 and


A) 21625

p units3 B) 64825

p units3 C) 64825 units3 D) 1944

25p units3

112)


113) The circle in which the plane through the point (10, 2, -3) perpendicular to the x-axis meets thesphere of radius 26 centered at the origin.

A) y 2 + z2 = 576 and x = 10 B) y 2 + z2 = 476

C) x2 + y 2 + z2 = 676 D) y 2 + z2 = 676 and x = 10

113)

Solve the problem.

114) Find the area of the triangle determined by the points P(-3, 6, -4), Q(2, -7, -3), and R(7, -6, -5).

A) 5 230 B) 9 4462

C) 9 446 D) 5 2302

114)


115) Let u = -7, -1 , v = 9, -7 . Find -5u + 6v.

A) -19, 47 B) 40, 12 C) 89, -37 D) -10, -48

115)


116) x2

32 + y

2

62 + z

2

32= 1


116)

16

Solve the problem.

117) A ramp leading to the entrance of a building is inclined upward at an angle of 4e. A suitcase is tobe pulled up the ramp by a handle that makes an angle of 39e with the horizontal. How much forcemust be applied in the direction of the handle so that the component of the force parallel to theramp is 50 lbs.?

A) 64.18 lbs B) 9.184 lbs C) 40.96 lbs D) -122.6 lbs

117)


118) y = 7 3x and y = 7 x


118)

119) y = 5 sin (x) and y = 5 cos (x)


119)


120) u = -8k, v = -7i

A) 8; -7j B) 56; 56j C) 56; -j D) 56; j

120)


121) The line through the point P(6, 1, -5) parallel to the vector 2i - 4j - 3k

A) x = 2t + 6, y = -4t + 1, z = -3t - 5 B) x = -2t - 6, y = 4t - 1, z = -3t + 5

C) x = 2t - 6, y = -4t - 1, z = -3t + 5 D) x = -2t+ 6, y = -4t + 1, z = 3t - 5

121)


122) u = 5i - 7j, v = j + k

A) u = - 72j - 72k + 5i - 21

2j - 72k B) u = - 7

2j - 72k + 5i - 7

2j + 72k

C) u = - 352

i + 492

j + - 5i - 472

j + k D) u = - 352

i + 492

j + - 352

i + 512

j + k

122)


123) 16i + 1

6j - 1

6k

A) 1216i + 1

6j - 1

6k B) 1

666i + 6

6j - 6

6k

C) 1336i + 3

6j - 3

6k D) 1

226i + 2

6j - 2

6k

123)


124) u = -5i - 5j, v = 2i + 6j + 8k

A) 1.58 B) 2.16 C) -0.59 D) 1.83

124)

17

Calculate the direction of P1P2 and the midpoint of line segment P1P2.

125) P1(4, 2, 7) and P2(6, 5, 1)

A) 27i + 3

7j - 6

7k; 5, 7

2, 4 B) 4

7i + 2

7j + 7

7k; 2, 1, 7

2

C) 27i + 37j - 67k; 5, 7

2, 4 D) 2

7i + 37j - 67k; 3, 5

2, 12

125)


126) u = -8i + 15j, v = -3j - 2k

A) u = 36013

i - 67513

j + - 36013

i + 63613

j - 45k B) u = 13513

j + 9013

k + - 8i + 33013

j + 9013

k

C) u = 36013

i - 67513

j + 36013

i - 71413

j - 45k D) u = 13513

j + 9013

k + - 8i + 6013

j - 9013

k

126)

Solve the problem.

127) Find a formula for the distance from the point P(x, y, z) to the z-axis.

A) x2 + y2 B) z + y C) x + y D) z2 + y2127)


128) The distance from the point S(-8, -10, -8) to the line x = -4 + 1t, y = -9 + 2t, z = -3 + 2t

A) 1223

B) 1223

C) 1229

D) 1229

128)

Solve the problem.

129) How much work does it take to slide a box 41 meters along the ground by pulling it with a 257 Nforce at an angle of 26e from the horizontal?

A) 6817 joules B) 9471 joules C) 231.0 joules D) 10,537 joules

129)

130) How much work does it take to slide a box 29 meters along the ground by pulling it with a 180 Nforce at an angle of 45e from the horizontal?

A) 2610 2 joules B) 5220 joules C) 52202 joules D) 5220 2 joules

130)


131) v = 3i - j + 3k, u = 11i + 2j + 10k

A) 67115

i + 12215

j + 1223

k B) 18319

i - 6119

j + 18319

k

C) 671225

i + 122225

j + 12245

k D) 19519

i - 6519

j + 19519

k

131)

18


132) z2

102 - x

2

102 - y

2100

= 1


132)

Solve the problem.

133) A bullet is fired with a muzzle velocity of 1106 ft/sec from a gun aimed at an angle of 28e above thehorizontal. Find the vertical component of the velocity.

A) 588.1 ft/sec B) 519.2 ft/sec C) -1065 ft/sec D) 976.5 ft/sec

133)

134) A bullet is fired with a muzzle velocity of 1163 ft/sec from a gun aimed at an angle of 32e above thehorizontal. Find the horizontal component of the velocity.

A) 970.2 ft/sec B) 726.7 ft/sec C) 986.3 ft/sec D) 616.3 ft/sec

134)


135) P1(3, 8, -5) and P2(7, 5, -10)

A) 225

i - 350

j - 110

k; 32, 4, - 5

2B) 452i - 3

52j - 1 2k; 7

2, 52, - 5

C) 225

i - 350

j - 110

k; 5, 132, - 152

D) 452i - 3

52j - 1 2k; 5, 13

2, - 152

135)


136) 2x2 + 2y2 + 2z2 - x + y - z = 9

A) C 14, - 14, 14, a = 5 6

2B) C 1

4, - 14, 14, a = 5 6

4

C) C - 14, 14, - 14, a = 75

8D) C - 1

4, 14, - 14, a = 5 3

4

136)

19

Solve the problem.

137) A bird flies from its nest 5 km in the direction 3e north of east, where it stops to rest on a tree. Itthen flies 8 km in the direction 4e south of west and lands atop a telephone pole. With anxy-coordinate system where the origin is the bird's nest, the x-axis points east, and the y-axispoints north, at what point is the telephone pole located?

A) (-7.981, -0.5581) B) (-2.987, -7.727) C) (4.993, 0.2617) D) (0.2792, 8.626)

137)


138) (1 + 3) x + (1 - 3) y = -4 and 3 x + y = -12

A) 45e B) 60e C) 75e D) 30e

138)


139) The plane through the point P(-7, 5, 6) and normal to n = 5i + 2j + 6k.

A) 5x + 2y + 6z = 11 B) 7x - 5y - 6z = 11

C) -7x + 5y + 6z = 11 D) -5x - 2y - 6z = 11

139)

Solve the problem.

140) Find the volume of the solid bounded by the paraboloid x249 + y

2100

= z5 and the planes z = 0 and


A) 448p2 units3 B) 896p units3 C) 448 units3 D) 448p units3

140)

Find v œœœœ u.

141) v = -4i + 9j and u = 6i + 9j

A) 2i + 18j B) -105 C) 57 D) -24i + 81j

141)


142) The plane through the point P(-6, 7, -6) and normal to n = -7i - 5j + 3k.

A) -6x + 7y + 6z = 95 B) 7x + 5y - 3z = 95

C) 6x - 7y - 6z = 95 D) -7x - 5y + 3z = -11

142)


143) x2 + y2 + z2 - 8x - 8y - 20z = -123

A) C(-4, -4, -10), a = 9 B) C(-4, -4, -10), a = 3

C) C(4, 4, 10), a = 9 D) C(4, 4, 10), a = 3

143)


144) P1(1, -2, -4) and P2(2, -3, -5)

A) 3 B) 3 C) 9 D) 2

144)


145) y2 + z2 = 3

A) Ellipsoid

B) Parabolic cylinder

C) Cylinder

D) Paraboloid

145)

20


146) The distance from the point S(-5, 5, 1) to the plane 3x + 4y = 4

A) 125

B) 15

C) 3125

D) 395

146)


147) x = 7, z = 3

A) All points in the x-z plane

B) The line through the point (7, 0, 3) and parallel to the y-axis

C) The line through the point (7, 3, 0) and parallel to the z-axis

D) The point (7, 3)

147)

Solve the problem.

148) Show that the point P(-5, 3, -6) is equidistant from the points A(-6, 1, -5) and B(-4, 5, -7).

A) The distance between P and A is 5; the distance between P and B is 5

B) The distance between P and A is 6; the distance between P and B is 6

C) The distance between P and A is 3; the distance between P and B is 3

D) The distance between P and A is 2; the distance between P and B is 2

148)

Find v œœœœ u.

149) v = 117, 117 and u = 1

17, -117

A) 217

B) 117

i - 117

j C) 217

i - 217

j D) 0

149)


150) x29 + z

29 = y2

A) Elliptic cone B) Paraboloid

C) Hyperbolic paraboloid D) Ellipsoid

150)


151) P = (-9, 5); perpendicular to v = -4i - 3j

A) -4x - 3y = 21 B) -4x - 3y = 25

C) -3x + 4y = 47 D) y - 5 = - 85(x + 4)

151)


152) The line through the point P(-1, 2, 0) and perpendicular to the plane 4x + 6y + 4z = 5

A) x = -6t - 1, y = -6t - 1, z = 0 B) x = 4t + 1, y = 6t - 2, z = 4t

C) x = 4t - 1, y = 6t + 2, z = 4t D) x = -4t + 1, y = -6t - 2, z = -4t

152)


153) The plane through the point P(1, -6, 2) and parallel to the plane -3x - 4y + 5z = 28.

A) -3x - 4y + 5z = 31 B) -3x - 4y + 5z = -11

C) -3x - 4y + 5z = -31 D) 4y = 31

153)

21

Solve the problem.

154) Find the work done by a force of 17i (newtons) in moving as object along a line from the origin tothe point (10, 3) (distance in meters).

A) 177.5 joules B) 17.75 joules C) 169.6 joules D) 170.0 joules

154)


155) The distance from the point S(3, -2, 2) to the line x = 2 + 11t, y = -1 + 2t, z = -6 + 10t

A) 692915

B) 13 4115

C) 13 41225

D) 6929225

155)


156) The half-space consisting of the points on and behind the yz-plane

A) x > 0 B) y > 0, z > 0 C) x ≤ 0 D) y ≤ 0, z ≤ 0

156)


157) u = -2i - 4i , v = i - j

A) 2; 2k B) 2; -2k C) 6; -6k D) 6; 6k

157)


158) P = (-5, 8); parallel to v = -2i - 9j

A) -9x + 2y = 61 B) y - 8 = - 173(x + 2)

C) -2x - 9y = -62 D) -2x - 9y = 85

158)

Solve the problem.

159) For the vectors u and v with magnitudes u = 6 and v = 7, find the angle q between u and vwhich makes proju v = 2

A) 70.53 B) 31.00 C) 19.47 D) 73.40

159)


160) (x - 5)2 + (y - 10)2 + (z - 4)2 < 16, - 4 ≤ z ≤ 0

A) All points within the lower hemisphere centered at (5, 10, 4)

B) All points on the lower hemisphere centered at (5, 10, 4)

C) No set of points satisfy the given relations.

D) All points outside the lower hemisphere centered at (5, 10, 4)

160)

Solve the problem.

161) A force of magnitude 6 pounds pulling on a suitcase makes an angle of 30e with the ground.Express the force in terms of its i and j components.

A) 0.8660i + + 0.5000j B) 3.000i + 5.196j

C) 5.196i + 3.000 j D) 0.9255i - 5.928j

161)


162) The distance from the point S(10, -2, 5) to the plane 10x + 11y + 2z = -10

A) 98225

B) 9815

C) 58225

D) 5815

162)

22


163) u = -7i - 11j + 1k, v = i - 4j - k

A) u = - 14i - 22j + 2k + - 13i - 26j - k B) u = 2i - 8j - 2k + - 9i - 3j + 3k

C) u = - 14i - 22j + 2k + 15i + 18j - k D) u = 2i - 8j - 2k + - 5i - 19j - 1k

163)


164) The rectangular solid in the first octant bounded by the planes x = 1, x = 4, y = 4, y = 5, z = 5 andz = 7 (planes excluded)

A) 1 < x < 4; 4 < y < 5; 5 < z < 7

B) x < y < z

C) The given planes do not form a rectangular solid

D) x < 1, x > 4; y < 4, y > 5; z < 5, z > 7

164)


165) x = 9 + 6t, y = 3 - 3t, z = 2 + 6t ; 10x + 7y - 4z = 2

A) - 1575, 1165, - 192

5B) 247

5, - 865, 66815

C) (3, 6, -4) D) (15, 0, 8)

165)


166) Let u = 3, 1 , v = 7, -5 . Find u - v.

A) 10, -4 B) 8, -6 C) -4, 6 D) 2, 12

166)


167) P1(-2, 0, -5) and P2(0, -5, 0)

A) x = -2t, y = 5t - 5, z = -5t B) x = 2t - 2, y = -5t, z = 5t - 5

C) x = 3t - 2, y = 6t, z = 6t - 5 D) x = 3t, y = 6t - 5, z = 6t

167)


168) The circle in which the plane through the point (2, 4, -7) perpendicular to the y-axis meets thesphere of radius 5 centered at the origin.

A) x 2 + z2 = 16 and y = 4 B) x 2 + z2 = 9 and y = 4

C) x 2 + z2 = -7 and y = 4 D) x 2 + z2 = 25 and y = 4

168)


169) The line through the point P(-7, 0, -6) and parallel to the line x = 3t - 6, y = 5t + 7, z = 3t - 6

A) x = 3t - 7, y = 5t, z = 3t - 6 B) x = -7, y = 3t, z = -5t - 6

C) x = 3 + 7, y = 5t, z = 3t + 6 D) x = 5t - 7, y = -3t, z = -6

169)


170) The slab bounded by the planes x = 2 and x = 3 (planes included)

A) x ≤ 2 and x ≥ 3 B) 2 ≤ x ≤ 3

C) -« < y < « and -« < z < « D) x = 2 and x = 3

170)

23

171) The closed region bounded by the spheres of radius 2 and 10, both centered at the origin, and theplanes x = 5 and x = 8

A) 4 < x2 + y2 + z2 < 100 and 5 < x < 8 B) 2 < x2 + y2 + z2 < 10 and x = 5 and x = 8

C) 2 ≤ x2 + y2 + z2 ≤ 10 and x = 5 and x = 8 D) 4 ≤ x2 + y2 + z2 ≤ 100 and 5 ≤ x ≤ 8

171)


172) x2 + (y + 7)2 + (z - 3)2 = 9

A) C(0, 7, -3), a = 3 B) C(0, -7, 3), a = 9 C) C(0, 7, -3), a = 9 D) C(0, -7, 3), a = 3

172)


173) P = (9, 10); parallel to v = 5i - 6j

A) -6x - 5y = -104 B) 5x - 6y = 61 C) y - 10 = 4(x - 5) D) 5x - 6y = -15

173)


174) The plane through the point A(-6, -2, 10) perpendicular to the vector from the origin to A.

A) 6x - 2y - 10z = -2 B) 6x + 2y - 10z = -140

C) -6x - 2y + 10z = -140 D) 6x + 2y - 10z = 140

174)


175) y2 + z2 = 25, x = -2

A) The circle y2 + z2 = 25 in the plane x = -2

B) All points more than 25 units from the origin

C) The line tangent to the circle y2 + z2 = 25 at the point x = -2

D) The cylinder with the radius 25 along the x-axis

175)


176) u = 2i - 3j - 10k, v = 3i + 4j - 5k

A) 0.95 B) 1.56 C) 0.63 D) 1.30

176)


177) y = 3 x + 16 and y = - 3 x + 9

A) 30e B) 60e C) 75e D) 45e

177)

24


178) y2

82 - x

2

82 = z3


178)

Solve the problem.

179) Find the perimeter of the triangle with vertices A(1, 2, 3), B(2, -2, 6), and C(3, 5, 7).

A) 98 + 51 + 113 B) 34 + 51 + 41

C) 10 + 51 + 69 D) 26 + 51 + 29

179)


180) u = -8j and v = 9i - 4k

A) 0.10 B) 0.00 C) 1.67 D) 1.57

180)


181) P1(-5, 7, 7) and P2(-4, 8, 8)

A) 13i + 13j + 13k; - 9

2, 152, 152

B) 53i + 7

3j + 7

3k; - 5

2, 72, 72

C) 13i + 1

3j + 1

3k; - 2, 4, 4 D) 1

3i + 1

3j + 1

3k; - 9

2, 152, 152

181)


182) P1P2 if P1 is the point (-1, -3, -1) and P2 is the point (1, -6, -5)

A) v = -2i - 3j + 4k B) v = -2i + 3j - 4k C) v = 2i - 3j - 4k D) v = -2i + 3j + 4k

182)


183) 9x - 8y = 6 and 6x + 5y = 6


183)

25

Find v œœœœ u.

184) v = 2i + 2j and u = 9i + 9j

A) 18i + 18j B) 11i + 11j C) 36 D) 0

184)


185) -2x - 3y - 5z = -2 and 5x + 4y - 7z = -1

A) 0.224 B) 1.347 C) 1.399 D) 0.972

185)


186) 13i - 1

6j - 1

2k

A) 23

32 3

i - 32 6

j - 32 2

k B) 1663i - 6

6j - 6

2k

C) 1 13i - 16j - 12k D) 1

223i - 2

6j - 2

2k

186)


187) The line through the point P(-2, -5, -5) parallel to the vector -2i + 5j - 6k

A) x = 2t + 2, y = 5t + 5, z = -6t + 5 B) x = -2t + 2, y =5t + 5, z = -6t + 5

C) x = -2t - 2, y = 5t - 5, z = -6t - 5 D) x = 2t - 2, y = 5t - 5, z = -6t - 5

187)


188) Center (-2, 1, 0), radius = 9

A) x2 + y2 + z2 + 4x - 2y = 76 B) x2 + y2 + z2 - 4x + 2y = 76

C) x2 + y2 + z2 + 4x + 2y = 76 D) x2 + y2 + z2 - 4x - 2y = 76

188)

Solve the problem.

189) A garden hose is spraying a stream of water at a box on the ground with a force of 2.6 pounds. Thewater stream makes an angle of 60 with the ground. What is the horizontal component of theforce?

A) 0.5000 B) -2.476 C) 1.300 D) 2.252

189)


190) P1(4, -1, -4) and P2(8, 3, -2)

A) 23i + 23j + 13k; 4, 3

2, - 1 B) 4

3i + 13j + 43k; 2, - 1

2, - 2

C) 23i + 23j + 13k; 2, 2, 1 D) 2

3i + 23j + 13k; 6, 1, - 3

190)


191) P1(4, 3, 4) and P2(0, 3, 7)

A) x = -4t + 4, y = 3t, z = 3t + 4 B) x = -4t + 4, y = 3, z = 3t + 4

C) x = 4t, y = 3t, z = -3t + 7 D) x = 4t, y = 3, z = -3t + 7

191)

26


192) u = -7i + 2i + 2k, v = 0

A) 0; 157(-7i + 2i + 2k) B) 0; 0

C) 0; -7i + 2i + 2k D) 57; 157(-7i + 2i + 2k)

192)

Solve the problem.

193) Let u = -2i + 8j, v = 2i + 3j, and w = i - j. Write u = u1 + u2 where u1 is parallel to v and u2 isparallel to w.

A) u1 = 3vu2 = 2w

B) u1 = -4vu2 = -24w

C) u1 = 0.8333 vu2 = -0.2273w

D) u1 = 1.200 vu2 = -4.400w

193)


194) y = x2 - 9 and y = 9 - x2


194)


195) The set of points equidistant from the points (0, 0, -8) and (0, 0, 1)

A) z > -8 and z < 1 B) x + y = 0 and -8 < z < 1

C) x + y = -3.5 D) z = -3.5

195)


196) The line through the point P(3, 2, -5) and perpendicular to the plane -3x + 6y + 5z = 5

A) x = -3t + 3, y = 6t + 2, z = 5t - 5 B) x = -3t - 3, y = 6t - 2, z = 5t + 5

C) x = 3t - 3, y = -6t - 2, z = -5t + 5 D) x = -6t + 3, y = -3t + 2, z = -5

196)


197) u = i - k, v = i + j

A) u = 12i - 12k + 1

2i + j - 1

2k B) u = 1

2i + 12k + 1

2i - j + 1

2k

C) u = 12i + 12j + 1

2i - 12j - k D) u = 1

2i + 12j + 1

2i - 12j + k

197)


198) x - 3 y = -2 and 3 x - y = 15

A) 60e B) 75e C) 30e D) 45e

198)


199) Let u = 2, -9 , v = -5, 4 . Find u + v.

A) 6, -14 B) -3, -5 C) 7, -13 D) -7, -1

199)

27


200) u = 5i - 8j + 3k, v = -2i + 4j - 4k

A) u = 3i - 6j + 6k + 2i - 2j - 3k

B) u = 3i - 6j + 6k + 8i - 14j + 9k

C) u = - 152

i + 12j - 92k + - 19

2i + 16j - 17

2k

D) u = - 152

i + 12j - 92k + 11

2i - 8j + 1

2k

200)


201) x2 + y2 + z2 = 49, z = 1

A) All points on the sphere x2 + y2 + z2 = 49 and above the plane z = 1

B) The sphere x2 + y2 + z2 = 1

C) All points within the sphere x2 + y2 + z2 = 49 and above the plane z = 1

D) The circle x2 + y2 = 48 in the plane z = 1

201)


202) x25 + y

29 = 7

A) Parabolic cylinder B) Paraboloid

C) Elliptical cylinder D) Ellipsoid

202)

203) x29 + y

23 = z

25

A) Hyperbolic paraboloid

B) Paraboloid

C) Ellipsoid

D) Elliptic cone

203)

Find v œœœœ u.

204) v = 2i - 3j and u = -6i + 7j

A) -33 B) 9 C) -12i - 21j D) -4i + 4j

204)

Solve the problem.

205) Find a unit vector perpendicular to plane PQR determined by the points P(3, -3, -1), Q(2, 1, -1),and R(2, -2, - 1).

A) ± 23i + 13j + 23k B) ±

29i + 19j + 29k

C) ± 29i + 29j + 19k D) ±

23i + 23j + 13k

205)


206) The plane through the points P(-1, -4, -5) , Q(2, -1, 13) and R(1, -7, 17).

A) 8x - 2y - z = -5 B) 8x + 2y + z = 5 C) 8x + 2y +z = -5 D) 8x - 2y - z = 5

206)

28


207) P1(-4, -8, -8) and P2(-2, -2, -11)

A) 7 B) 7 C) 5 D) 6

207)

Solve the problem.

208) Find the work done by a force of 9i (newtons) in moving as object along a line from the origin to thepoint (10, 10) (distance in meters).

A) 63.64 joules B) 90 joules C) 90 2 joules D) 9 2 joules

208)


209) x2 + y2 = 16


209)


210) The vector from the point A(8, 5) to the origin

A) (-8, -5) B) (-8, 5) C) (8, 5) D) (8, -5)

210)


211) u = -5i - 3j + 4j; v = 4i - 8j + 8k; w = 6i - 5j - 9k

A) -868 B) 32 C) -796 D) -196

211)

Solve the problem.

212) A bullet is fired with a muzzle velocity of 1438 ft/sec from a gun aimed at an angle of 19e above thehorizontal. Find the horizontal component of the velocity.

A) 1360 ft/sec B) 495.1 ft/sec C) 1422 ft/sec D) 468.2 ft/sec

212)

29


213) 3i - 92j - 1k

A) 112611

i - 911

j - 211

k B) 112

6121

i - 9121

j - 2121

k

C) 1123i - 9

2j - 1k D) 11

2(i - j - k)

213)


214) Let u = 4, 9 . Find 7u.

A) 28, 63 B) -28, 63 C) 28, -63 D) -28, -63

214)


215) y = sin (4x) and y = 8x


215)

Solve the problem.

216) An airplane is flying in the direction 72e west of north at 765 km/hr. Find the component form ofthe velocity of the airplane, assuming that the positive x-axis represents due east and the positivey-axis represents due north.

A) -0.9511, 0.3090 B) -236.4, 727.6 C) -194.2, -739.9 D) -727.6, 236.4

216)


217) 8x + 3y + 4z = -3 and 2x + 9y + 8z = -1

A) 0.862 B) 0.709 C) 1.440 D) 1.189

217)


218) The plane through the point P(-10, 8, -5) and perpendicular to the line x = 9 + 9t, y = 6 + 6t,z = 4 + 6t

A) 9x + 6y + 6z = -72 B) 9x + 6y + 6z = 21

C) 9x + 6y + 6z = 72 D) 9x + 6y + 6z = -7

218)


219) z210 - x

29 = y6

A) Ellipsoid B) Hyperbolic paraboloid

C) Paraboloid D) Parabolic cylinder

219)


220) 65j - 85k

A) 2(j - k) B) 2 65j - 85k C) 2 3

5j - 45k D) 4 3

5j - 45k

220)


221) y = 6x2 and y = 9x3


221)

30


222) P = (9, 3); parallel to v = 2i + 5j

A) y - 3 = - 27(x - 2) B) 2x + 5y = 29 C) 2x + 5y = 33 D) 5x - 2y = 39

222)


223) 5 ≤ y ≤ 6, 5 ≤ z ≤ 6

A) The infinitely long square prism parallel to the x-axis

B) The line between the points (0, 5, 5) and (0, 6, 6)

C) The cube located in the first quadrant and with sides 5 units in length

D) The square with corners at (0, 5, 5), (0, 5, 6), (0, 6, 5), and (0, 6, 6)

223)


224) Center (0, -3, -1), radius = 2

A) x2 + y2 + 2z2 + 6y + 2z = -6 B) x2 + y2 + z2 + 6y + 2z = -6

C) x2 + y2 + 2z2 + 6y - 2z = -6 D) x2 + y2 + z2 + 6y - 2z = -6

224)


225) The interior of the sphere of radius 4 centered at the point (4, 2, 1)

A) (x - 4)2 + (x - 2)2 + (x - 1)2 > 16 B) (x + 4)2 + (x + 2)2 + (x + 1)2 ≥ 16

C) (x - 4)2 + (x - 2)2 + (x - 1)2 < 16 D) (x + 4)2 + (x + 2)2 + (x + 1)2 > 16

225)


226) x28 + y

25 - z

25 = 1

A) Hyperboloid of one sheet B) Elliptic cone

C) Hyperboloid of two sheets D) Ellipsoid

226)


227) -8 ≤ z ≤ -7

A) No set of points satisfies the given relation

B) All points between z = -8 and z = -7 in the x-y plane

C) The slab between the planes z = -8 and z = -7 (including planes)

D) The line from z = -8 to z = -7

227)


228) (x + 4)2 + (y + 7)2 + (z + 4)2 = 149

A) C(-4, -7, -4), a = 17

B) C(4, 7, 4), a = 149

C) C(4, 7, 4), a = 17

D) C(-4, -7, -4), a = 149

228)

31

Solve the problem.

229) Find the magnitude of the torque in foot-pounds at point P for the following lever:

PQ = 10 in. and F = 25 lb

A) 250 ft-lb B) -12,188.30 ft-lb C) 1706.36 ft-lb D) 12,188.30 ft-lb

229)


230) Let u = 4, -2 , v = -6, -7 . Find 513

u - 1213

v.

A) 7413, 9213

B) 4413, 5413

C) 5013, 6031

D) 9213, 7413

230)


231) x2 + y2 + z2 > 49

A) All points outside the sphere of radius 7

B) All points on the surface of the cylinder with radius 7

C) All points outside the cylinder with radius 7

D) All points in space

231)


232) y = e2x and y = e8x


232)


233) -8x + 7y + 9z = -9, 5x - 4y + 8z = 7

A) x = -276t + 13, y = -327t + 11, z = 3t B) x = -92t - 13, y = -109t + -11, z = 3t

C) x = 92t + 133, y = 109t + 11

3, z = -3t D) x = 92t + 13, y = 109t - 11, z = 3t

233)


234) 4u - 3v if u = 1, 1, 0 and v = 3, 0, 1

A) v = 13i + 4j - 3k B) v = -5i + 7j - 3k C) v = -5i + 4j - 3k D) v = 4i + 4j - 3k

234)


235) The distance from the point S(-3, -6, 2) to the plane -9x + 2y + 6z = 10

A) 1711

B) 17121

C) 13121

D) 1311

235)

32


236) Center (0, 0, 5), radius = 6

A) x2 + y2 + z2 - 10z = 11 B) x2 + y2 + 6 z2 - 10z = 11

C) x2 + y2 + z2 + 10z = 11 D) x2 + y2 + 6 z2 + 10z = 11

236)


237) u = 3j - 6k, v = 6i - 9j - 6k

A) 1.46 B) 1.53 C) 0.11 D) 1.57

237)


238) The plane through the point (5, -4, -4) and parallel to the xy-plane

A) x + y - 8 = 0 B) z = -8 C) x + y = 15 D) x = 5 and y = 10

238)


239) x2

62 + y

2

62 = z

236


239)

Solve the problem.

240) For the triangle with vertices located at A(4, 3, 3), B(2, 5, 2), and C(1, 1, 1) , find a vector from vertexC to the midpoint of side AB.

A) 2i + 3j + 32k B) 1i + 1

2j + 12k C) 3i + 4j + 5

2k D) 4i + 5j + 7

2k

240)

241) Find a unit vector perpendicular to plane PQR determined by the points P(1, -1, -1), Q(-1, 1, 53),

and R(-2, 4, 5).

A) ± -6121

i + 2121

j + 6121

k B) ± -6121

i - 2121

j - 6121

k

C) ± -611

i - 211

j - 611

k D) ± -611

i + 211

j + 611

k

241)

33


242) 6j

A) 6(6j) B) 16j C) 6j D) 6 1

6j

242)


243) u = -8i + 6j - 8k, v = 7i + 6j - 9k

A) 1.57 B) 1.25 C) 0.32 D) 1.41

243)


244) v = 3j, u = 4i + 3k

A) 0 B) 49i + 1

3j C) 3

10j D) 4

9i + 1

3j + 1

3k

244)

34

Answer KeyTestname: TEST 3

1) Verify that a + b2

= a - b2

.

Cancel the 2's and square both sides:a + b 2 = a - b 2 or(a + b)œ(a + b) = (a - b)œ(a - b) oraœa + 2aœb + bœb = aœa - 2aœb + bœb [2aœb = 0 since a and b are orthogonal]aœa + bœb = aœa + bœb Verified.Thus, the midpoint is equidistant from all three vertices.

2) Not always true; The statement is false if u ≠ v.3)

4) Always true by distributive property5) AœB = (au + bv)œ(bu - av) = abuœu - a2uœv + b2uœv - abvœv = ab - ab = 0Since the dot product is zero and since neither A nor B is identically zero, A and B are orthogonal.

6) Not always true; (u ˛ v) œ w = u œ (v ˛ w), but v ˛ w = -(w ˛ v) from which it follows that the original equation false ifw ˛ v ≠ 0.

35


7)

8) Let w = xi + yj. Then the dot product is wœ v = w v cosq ≥ 0. The relation is satisfied for all q between - p

2 and p

2.

9) Always true because u ˛ v and v are orthogonal10)

36


11)

12) Not always true; The statement if false if c ≠ 0,1.13)

14)

37


15)

16)

17) Always true by definition of the cross product18) Take the dot product:

( a b + b a) œ ( a b - b a) = ( a b + b a) œ a b - ( a b + b a) œ b a= ( a b) œ ( a b) + ( b a) œ ( a b) - ( a b) œ ( b a) - ( b a) œ ( b a)= a 2bœb + a b a œ b - a b a œ b - b 2 aœa= a 2 b 2 - b 2 a 2 = 0

Since the dot product is zero and since neither vector is identically zero, then the vectors are orthogonal.

38


19)

20) Always true by definition21) Not always true; The statement if false if c ≠ 0,1.22) u = ux i + uyj and v = vx i + vyj , so

a = u + v = (ux + vx) i + (uy + vy)j and b = u - v = (ux - vx) i + (uy - vy)j

Take the dot product a œ b:a œ b = (u + v) œ (u - v) = (ux + vx)(ux - vx) + (uy + vy)(uy - vy)

= u 2x - v2x + u

2y - v

2y = ( u

2x + u

2y ) - ( v

2x + v

2y )

= u - v = 1 - 1 = 0Since the dot product of the two non-zero vectors is zero they are orthogonol.

23) Always true by definition of 024)

25) D26) D27) A28) B29) C30) C

39


31) D32) A33) A34) A35) A36) B37) C38) C39) C40) C41) C42) B43) C44) D45) C46) D47) A48) C49) D50) D51) A52) A53) D54) C55) C56) B57) C58) A59) A60) C61) D62) C63) B64) A65) D66) B67) B68) C69) C70) A71) C72) D73) A74) B75) A76) D77) B78) A79) D80) C

40


81) B82) B83) B84) D85) D86) C87) C88) A89) B90) B91) D92) A93) C94) A95) A96) B97) B98) D99) C100) D101) B102) D103) A104) D105) B106) B107) C108) A109) D110) C111) D112) B113) A114) D115) C116) C117) A118) C119) C120) D121) A122) B123) D124) B125) C126) D127) A128) A129) B130) A

41


131) B132) A133) B134) C135) D136) D137) B138) A139) A140) D141) C142) D143) D144) A145) C146) B147) B148) B149) D150) B151) A152) C153) A154) D155) B156) C157) C158) A159) A160) A161) C162) B163) B164) A165) A166) C167) B168) B169) A170) B171) D172) D173) A174) B175) A176) A177) B178) A179) D180) B

42


181) D182) C183) C184) C185) B186) C187) C188) A189) C190) D191) B192) B193) D194) A195) D196) A197) C198) C199) B200) A201) D202) C203) D204) A205) D206) D207) B208) B209) C210) A211) C212) A213) A214) A215) A216) D217) A218) A219) B220) C221) D222) D223) A224) B225) C226) A227) C228) A229) D230) D

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231) A232) D233) C234) C235) A236) A237) A238) B239) A240) A241) C242) C243) B244) A

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Exam - Okan Üniversitesiusers.okan.edu.tr/sezgin.sezer/public_html/Test 3.pdf · 2015-11-24 · 11)z = x2 + 4y2 11) Determine whether the following is always true or not always true