Deck 13: Vectors and the Geometry of Space

Question 1

Find the indicated vector.

-Let u = , v = . Find u + v.

A)
B)
C)
D)

Answer 1

Question 2

Find the indicated vector.

-Let u = , v = . Find u - v.

A)
B)
C)
D)

Answer 2

Question 3

Find the indicated vector.

-Let u = , v = . Find v - u.

A)
B)
C)
D)

Answer 3

Question 4

Find the indicated vector.

-Let u = . Find 8u.

A)
B)
C)
D)

Question 5

Find the indicated vector.

-Let u = . Find 2u.

A)
B)
C)
D)

Question 6

Find the indicated vector.

-Let u = . Find -8u.

A)
B)
C)
D)

Question 7

Find the indicated vector.

-Let u = . Find -6u.

A)
B)
C)
D)

Question 8

Find the indicated vector.

-Let u = , v = . Find -u + 2v.

A)
B)
C)
D)

Question 9

Find the indicated vector.

-Let u = , v = . Find u + v.

A)
B)
C)
D)

Question 10

Find the indicated vector.

-Let u = , v = . Find u - v.

A)
B)
C)
D)

Question 11

Find the corresponding position vector.

-Define the points P = ( -10, -6) and Q = ( -2, 5). Find the position vector corresponding to .

A)
B)
C)
D)

Question 12

Express the vector in the form ai + bj.

- if is the point ( 2, -6) and is the point ( 4, -9)

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

Question 13

Express the vector in the form ai + bj.

- if A is the point ( -1, -7) and B is the point ( 4, -14)

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

Question 14

Solve the problem.

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

A) = 3v
= 2w
B) = -4 v
= -24 w
C) = 1.200 v
= -4.400 w
D) = 0.8333 v
= - 0.2273 w

Question 15

Solve the problem.

-A force of magnitude 8 pounds pulling on a suitcase makes an angle of 30° with the ground. Express the force in terms of its i and j components.

A) 6.928i + 4.000 j
B) 4.000i + 6.928j
C) 1.234i - 7.904j
D) 0.8660i + + 0.5000j

Question 16

Solve the problem.

-An airplane is flying in the direction 50° west of north at 817 km/hr. Find the component form of the velocity of the airplane, assuming that the positive x-axis represents due east and the positive y-axis represents due north.

A)
B)
C)
D)

Question 17

Solve the problem.

-An airplane is flying in the direction 10° east of south at 701 km/hr. Find the component form of the velocity of the airplane, assuming that the positive x-axis represents due east and the positive y-axis represents due north.

A)
B)
C)
D)

Question 18

Solve the problem.

-A bird flies from its nest 2 km in the direction 39° north of east, where it stops to rest on a tree. It then flies 5 km in the direction 13° south of west and lands atop a telephone pole. With an xy-coordinate system where the origin is the bird's nest, the x-axis points east, and the y-axis points north, at what point is the tree located?

A) ( 1.259, 1.554)
B) ( 1.554, 1.259)
C) ( 0.5333, 1.928)
D) ( 0.7771, 0.6293)

Question 19

Solve the problem.

-A bird flies from its nest 4 km in the direction 4° north of east, where it stops to rest on a tree. It then flies 7 km in the direction 4° south of west and lands atop a telephone pole. With an xy-coordinate system where the origin is the bird's nest, the x-axis points east, and the y-axis points north, at what point is the telephone pole located?

A) ( -6.983, -0.4883)
B) ( 3.990, 0.2790)
C) ( -2.993, -0.2093)
D) ( 1.961, 1.548)

Question 20

Solve the problem.

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

A)

B)

C)

D)

Question 21

Express the vector in the form ai + bj + ck.

- if is the point ( 3, 6, -4) and is the point ( 5, 3, -8)

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

Question 22

Express the vector in the form ai + bj + ck.

- if A is the point ( -4, -8, -5) and B is the point ( 1, -15, -2)

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

Question 23

Express the vector in the form ai + bj + ck.

-2u - 6v if u = and v =

A) v = 2i + 2j - 6k
B) v = -16i + 8j - 6k
C) v = 20i + 2j - 6k
D) v = -16i + 2j - 6k

Question 24

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

-Center (0, -10, -7), radius = 5

A)

B)

C)

D)

Question 25

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

-Center (-8, 10, 0), radius = 5

A)

B)

C)

D)

Question 26

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

-Center (0, 0, 5), radius = 10

A)

B)

C)

D)

Question 27

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

-Center (-2, 0, 0), radius = 4

A)

B)

C)

D)

Question 28

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

- + + > 1

A) All points outside the sphere of radius 1
B) All points in space
C) All points outside the cylinder with radius 1
D) All points on the surface of the cylinder with radius 1

Question 29

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

- + + < 1

A) All points outside the lower hemisphere centered at ( 1, 7, 3)
B) All points on the lower hemisphere centered at ( 1, 7, 3)
C) No set of points satisfy the given relations.
D) All points within the lower hemisphere centered at ( 1, 7, 3)

Question 30

Find v ? u.

-v = 5i + 4j and u = 2i + 6j

A) 34
B) 7i + 10j
C) -14
D) 10i + 24j

Question 31

Find v ? u.

-v = -3i + 6j and u = 9i + 3j

A) -27i + 18j
B) 6i + 9j
C) -45
D) -9

Question 32

Find v ? u.

-v = 3i - 6j and u = -9i + 2j

A) -6i - 4j
B) -39
C) -27i - 12j
D) -15

Question 33

Find v ? u.

-v = 8i - 7j and u = -4i - 5j

A) 4i - 12j
B) 3
C) -67
D) -32i + 35j

Question 34

Find v ? u.

-v = 3i - 7j and u = -3 i + 7j

A) ( 3 - 3 )i + 0j
B) -49 - 9
C) -49 i - 9j
D) -49 + 9

Question 35

Find v ? u.

-v = and u =

A) i - j
B) 0
C)
D) i - j

Question 36

Find v ? u.

-v = and u =

A) i - j
B) i
C)
D) 0

Question 37

Find the angle between u and v in radians.

-u = 9i + 4j + 7k, v = 9i + 2j + 8k

A) 1.39
B) 1.06
C) 1.49
D) 0.18

Question 38

Find the angle between u and v in radians.

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

A) 1.57
B) 0.50
C) 1.33
D) 1.08

Question 39

Find the angle between u and v in radians.

-u = 4i - 9j - 6k, v = 5i + 5j - 5k

A) 1.57
B) 1.55
C) 0.05
D) 1.52

Question 40

Find the angle between u and v in radians.

-u = -2i - 2j, v = 4i + 9j + 8k

A) 1.59
B) 1.73
C) -0.81
D) 2.38

Question 41

Find the angle between u and v in radians.

-u = 7j - 2k, v = 4i - 6j - 7k

A) -0.39
B) 1.75
C) 1.96
D) 1.58

Question 42

Find the angle between u and v in radians.

-u = 8i, v = 7i - 9j

A) 0.91
B) 0.66
C) 1.28
D) 1.56

Question 43

Find the angle between u and v in radians.

-u = -10j and v = 6i - 8k

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

Question 44

Find u.

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

A)

B)

C)

D)

Question 45

Find u.

-v = i + j + k, u = 12i + 3j + 4k

A)

B)

C)

D)

Question 46

Find u.

-v = k, u = 6i + 9j + 2k

A)

B)

C)

D)2k

Question 47

Find u.

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

A)

B)

C)

D)

Question 48

Find u.

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

A)

B)

C)

D)

Question 49

Find u.

-v = 3j, u = 3i + 4k

A)

B)

C)

D) 0

Question 50

Solve the problem.

-A bullet is fired with a muzzle velocity of 1159 ft/sec from a gun aimed at an angle of 29° above the horizontal. Find the horizontal component of the velocity.

A) 1014 ft/sec
B) -867.0 ft/sec
C) 561.9 ft/sec
D) 642.4 ft/sec

Question 51

Solve the problem.

-A bullet is fired with a muzzle velocity of 1489 ft/sec from a gun aimed at an angle of 12° above the horizontal. Find the horizontal component of the velocity.

A) 309.6 ft/sec
B) 1256 ft/sec
C) 1456 ft/sec
D) 316.5 ft/sec

Question 52

Solve the problem.

-A ramp leading to the entrance of a building is inclined upward at an angle of 6°. A suitcase is to be pulled up the ramp by a handle that makes an angle of 34° with the horizontal. How much force must be applied in the direction of the handle so that the component of the force parallel to the ramp is 50 lbs.?

A) 44.15 lbs
B) 59.98 lbs
C) -56.58 lbs
D) 55.90 lbs

Question 53

Solve the problem.

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

A) 38.18 joules
B) 54 joules
C) 9 joules
D) 54 joules

Question 54

Solve the problem.

-Find the work done by a force of 19i (newtons) in moving an object along a line from the origin to the point (distance in meters).

A) 28.60 joules
B) 152.0 joules
C) 228.8 joules
D) 98.65 joules

Question 55

Solve the problem.

-How much work does it take to slide a box 37 meters along the ground by pulling it with a 213 N force at an angle of 30° from the horizontal?

A) 3940.5 joules
B) 7881 joules
C) joules
D) 7881 joules

Question 56

Solve the problem.

-How much work does it take to slide a box 6 meters along the ground by pulling it with a 158 N force at an angle of 15° from the horizontal?

A) 915.7 joules
B) -720.2 joules
C) 152.6 joules
D) 948 joules

Question 57

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

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

Question 58

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

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

Question 59

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

-Show that the midpoint of the hypotenuse of a right triangle is equidistant from all three vertices. [Hint: See the figure below. Show that = .]

Question 60

Find the magnitude of u × v and the unit vector parallel to u × v in the direction of u × v.

-u = 4i, v = 3j

A) 4; 3k
B) 12; -k
C) 12; 12k
D) 12; k

Question 61

Find the magnitude of u × v and the unit vector parallel to u × v in the direction of u × v.

-u = 7i + 5j , v = i - j

A) 2; 2k
B) 2; -2k
C) 12; -k
D) 12; -12k

Question 62

Find the magnitude of u × v and the unit vector parallel to u × v in the direction of u × v.

-u = -3i - 2j - 3k, v = 6i + 4j + 6k

A) ; ( 3 i + 2 j + 3 k )
B) 0; no direction
C) 0; 3i+ 2j + 3k
D) ; ( 3 i + 2 j + 3 k )

Question 63

Find the magnitude of u × v and the unit vector parallel to u × v in the direction of u × v.

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

A) 180; i + k
B) 6 ; i + k
C) 180; i + j + k
D) 6 ; i - k

Question 64

Find the magnitude of u × v and the unit vector parallel to u × v in the direction of u × v.

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

A) 3; i + j - k
B) 9; i + j - k
C) 3; i - j + k
D) 9; i - j + k

Question 65

Find the magnitude of u × v and the unit vector parallel to u × v in the direction of u × v.

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

A) 2 ; - i + j - k
B) 8; i - j - k
C) 2 ; i + j - k
D) 8; i - j + k

Question 66

Find the magnitude of u × v and the unit vector parallel to u × v in the direction of u × v.

-u = -3i - 8i + 5k, v = 0

A) 0; no direction
B) 0; -3i - 8i + 5k
C) 7 ; ( -3i - 8i + 5k)
D) 0; ( -3i - 8i + 5k)

Question 67

Solve the problem.

-Find the area of the triangle determined by the points P(1, 1, 1), Q( 6, 6, -3), and

A)
B)
C)
D)

Question 68

Solve the problem.

-Find the area of the triangle determined by the points P(-3, 6, -4), Q( 10, -1, -2), and

A)
B)
C)
D) 3

Question 69

Solve the problem.

-Find the area of the parallelogram determined by the points P(7, -5, 5), Q( 2, -7, -3), and

A)
B)
C)
D)

Question 70

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

-u = i + j + k; v = 8i + 5j + 2k; w = 9i + 6j + 4k

A) -9
B) 25
C) 75
D) -3

Question 71

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

-u = -5i - 3j + 4k; v = 5i - 4j + 2k; w = 10i - 5j - 8k

A) -330
B) -3
C) -410
D) 30

Question 72

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

-u = 2i - 4j + 3k; v = -5i - 7j + 8k; w = 9i - 2j + 4k

A) 563
B) -39
C) 153
D) -173

Question 73

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

-u = 4i + 2j - k; v = 6i + 6j - 4k; w = 1i + 8j - 4k

A) 70
B) -50
C) -222
D) 30

Question 74

Solve the problem.

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

A) ft-lb
B) ft-lb
C) - ft-lb
D) ft-lb

Question 75

Solve the problem.

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

A) 6.34 ft-lb
B) 3.78 ft-lb
C) 50 ft-lb
D) 1.76 ft-lb

Question 76

Solve the problem.

-For the vectors u and v with magnitudes = 7 and = 8, find the angle $\theta$ between u and v which makes = 5

A) 45.58°
B) 44.42°
C) 28.96°
D) 51.32°

Question 77

Determine whether the following is always true or not always true. Given reasons for your answers.
-|u| =

Question 78

Determine whether the following is always true or not always true. Given reasons for your answers.
-u × 0 = 0

Question 79

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

-(u × v) . w = u . (w × v)

Question 80

Determine whether the following is always true or not always true. Given reasons for your answers.
-u × v = -(v × u)