Deck 11: Vectors and the Geometry of Space

Question 1

Identify the following quadric surface.

A) hyperboloid of two sheets
B) elliptic cone
C) ellipsoid
D) elliptic paraboloid
E) hyperboloid of one sheet

Answer 1

E

Question 2

Find the magnitude of the vector v given its initial and terminal points. Round your answer to four decimal places. Initial point: (3, -3, -4)
Terminal point: (8, -7, -1)

A) or 7.2801
B) or 7.6158
C) or 7.0711
D) or 4.3589
E) or 7.8102

Answer 2

C

Question 3

Identify the equation (written in terms of cylindrical or spherical coordinates) of the following graph. ​ ​

A)
B)
C)
D)
E)

Answer 3

C

Question 4

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

A)
B)
C)
D)
E)

Question 5

Find the standard equation of a sphere that has diameter with the end points given below. (4, 8, 3), (2, 4, 7)

A)
B)
C)
D)
E)

Question 6

Find the magnitude of the vector given below. Round your answer to four decimal places.

A)
B)
C)
D)
E)

Question 7

Find the vector v with the magnitude and direction of . ​
​

A)
B)
C)
D)
E)

Question 8

Complete the square to write the following equation in the standard form for a sphere. ​ ​

A)
B)
C)
D)
E)

Question 9

A chute at the top of a grain elevator of a combine funnels the grain into a bin (see figure). Find the angle between two adjacent sides. Round your answer to one decimal place. ​

A) 83.7
B) 91.6
C) 90.7
D) 86.4
E) 181.6

Question 10

Find the distance between the points given below. ​
(3, 1, 4 ), ( 6, 6, 7 )
​

A) or 6.5574 to four decimal places.
B) or 6.4807 to four decimal places.
C) or 6.6332 to four decimal places.
D) or 6.0828 to four decimal places.
E) or 6.7082 to four decimal places.

Question 11

Identify the equation of the following graph.

A)
B)
C)
D)
E)

Question 12

Sketch the graph of the cross product of the unit vector . ​

A)
B)
C)
D)
E)

Question 13

Find the standard equation of the sphere with center (4, 1, 1), and radius 2.

A)
B)
C)
D)
E)

Question 14

Identify the equation (written in terms of cylindrical or spherical coordinates) of the following graph. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 15

Graph the solid described in spherical coordinates. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 16

Determine whether u and v are orthogonal, parallel and pointing in the same direction, parallel and pointing in opposite directions, or neither.

A) Neither parallel nor orthogonal
B) Parallel, same direction
C) Orthogonal
D) Parallel, opposite direction

Question 17

Find the component form of the vector u with the given initial and terminal points. Initial point: (2, 1, 1 )
Terminal point: (7 , 10 , 3 )

A)
B)
C)
D)
E)

Question 18

Find a set of parametric equations of the line that passes through the point and is parallel to the xy-plane and yz-plane.

A)
B)
C)
D)
E)

Question 19

Find the standard equation of the sphere with center at and tangent to the yz-plane.

A)
B)
C)
D)
E)

Question 20

Identify the equation of the following graph.

A)
B)
C)
D)
E)

Question 21

The vector v and its initial point is given. Find the terminal point. , initial point

A)
B)
C)
D)
E)

Question 22

The top of a rubber bushing designed to absorb vibrations in an automobile is the surface of revolution generated by revolving the curve in the yz-plane about the z-axis. The bushing has a hole of 1 centimeter in diameter through its center and parallel to the axis of revolution. All measurements are in centimeters and the bushing is set on the xy-plane. Find the volume of the rubber bushing. Round your answer to two decimal places. ​

A)
B)
C)
D)
E)

Question 23

To carry a 100-pound cylindrical weight, two workers lift on the ends of short ropes tied to an eyelet on the top center of the cylinder. One rope makes a angle away from the vertical and the other makes a angle as shown in the figure below. Find each rope's tension if the resultant force is vertical.Round your answer to two decimal places. ​ ​

A)
B)
C)
D)
E)

Question 24

Find a unit vector that is orthogonal to both and .

A)
B)
C)
D)
E)

Question 25

Find the distance between the point and the plane .

A)
B)
C)
D)
E)

Question 26

Find the triple scalar product .

A)
B)
C)
D)
E)

Question 27

Suppose a car is towed using a force of 1300 newtons. The chain used to pull the car makes a angle with the horizontal. Find the work done in towing the car 3 kilometers. Round your answer to two decimal places.

A) 7,070.20 km-N
B) 3,534.60 km-N
C) 3,549.60 km-N
D) 7,068.20 km-N
E) 1,648.21 km-N

Question 28

Find the area of a parallelogram that has the given vectors as adjacent sides. Round your answer to two decimal places.

A)
B)
C)
D)
E)

Question 29

Given and , find . ​

A)
B)
C)
D)
E)

Question 30

Three forces with magnitudes pounds, pounds and pounds act on an object at angles , , and respectively, with the positive x-axis. Find the direction and magnitude of the resultant force. (The choices below are given to two decimal places.)

A) pounds at an angle of with the positive x-axis
B) pounds at an angle of with the positive x-axis
C) pounds at an angle of with the positive x-axis
D) pounds at an angle of with the positive x-axis
E) pounds at an angle of with the positive x-axis

Question 31

Identify the following quadric surface.

A) hyperboloid of one sheet
B) elliptic cone
C) elliptic paraboloid
D) hyperboloid of two sheets
E) hyperbolic paraboloid

Question 32

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

A)
B)
C)
D)
E)

Question 33

The top of a rubber bushing designed to absorb vibrations in an automobile is the surface of revolution generated by revolving the curve in the yz-plane about the z-axis. All measurements are in centimeters and the bushing is set on the xy-plane. Use the shell method to find its volume. Round your answer to one decimal place. ​

A)
B)
C)
D)
E)

Question 34

Find the set of parametric equations of the line through the point that is parallel to the line and . ​

A)
B)
C)
D)
E)

Question 35

The top of a rubber bushing designed to absorb vibrations in an automobile is the surface of revolution generated by revolving the curve in the yz-plane about the z-axis. Find an equation for the surface of revolution.

A)
B)
C)
D)
E)

Question 36

Some planet is an oblate ellipsoid rather than a sphere because of the forces caused by its rotation. The equatorial radius of this planet is 4851 miles and the polar radius is 4660 miles. Find the equation of the ellipsoid. (Assume that the center of a planet is at the origin and that the trace formed by the plane corresponds to the equator.)

A)
B)
C)
D)
E)

Question 37

Two insects are crawling along different lines in three-space. At time t (in minutes), the first insect is at the point where . Also, at time t, the second insect is at the point where . Assume distances are given in inches. Find the distance between the two insects at time . Round your answer to two decimal places.

A) inches
B) inches
C) inches
D) inches
E) inches

Question 38

Identify the following quadric surface.

A) hyperboloid of one sheet
B) elliptic cone
C) hyperboloid of two sheets
D) elliptic paraboloid
E) ellipsoid

Question 39

Suppose a plane is flying in the direction (relative to the west). Its speed with respect to the air is 600 kilometers per hour. The wind at the plane's altitude is from the southwest at 80 kilometers per hour (see figure). What is its speed with respect to the ground? Round your answer to one decimal place. ​ ​

A)
B)
C)
D)
E)

Question 40

Given , find .

A)
B)
C)
D)
E)

Question 41

Find vectors u and v whose initial and terminal points are given. Determine whether u and v are equivalent. ​
U: (4,2), (7,6) v: (-2,3), (1,7)
​

A) , which are equivalent
B) , which are not equivalent
C) , which are equivalent
D) , which are equivalent
E) , which are equivalent

Question 42

Convert the point from rectangular coordinates to spherical coordinates.

A)
B)
C)
D)
E)

Question 43

Find the equation of the surface satisfying the conditions, and identify the surface. The set of points equidistant from the point (-4, 3, 2 ) and the plane .

A) , an elliptic paraboloid
B) , a sphere
C) , a hyperbolic paraboloid
D) , a sphere
E) , a hyperbolic paraboloid

Question 44

An object is pulled 5 feet horizontally across a floor, using a force of 65 pounds. The direction of the force is above the horizontal. Find the work done. Round your answer to two decimal places.

A) 294.55 ft-lb
B) 322.14 ft-lb
C) 137.35 ft-lb
D) 5.62 ft-lb
E) 336.74 ft-lb

Question 45

Find a set of symmetric equations of the line through the points and .

A)
B)
C)
D)
E)

Question 46

Convert the point from spherical coordinates to rectangular coordinates.

A)
B)
C)
D)
E)

Question 47

Suppose a plane is flying in the direction . Its speed with respect to the air is 900 kilometers per hour. The wind at the plane's altitude is from the southwest at 100 kilometers per hour (see figure). What is the true direction of the plane? Round your answer to two decimal places. ​ ​

A) North of West
B) North of West
C) North of East
D) North of West
E) North of South

Question 48

Find the area of the parallelogram with the vertices , and . Round your answer to two decimal places.

A) 122.00
B) 6.00
C) 11.05
D) 12.00
E) 3.46

Question 49

Find the direction cosines of the vector u given below.

A)
B)
C)
D)
E)

Question 50

Find the volume of the parallelepiped with the following vertices.

A)
B)
C)
D)
E)

Question 51

Given and find .

A)
B)
C)
D)
E)

Question 52

Suppose a plane flies at a constant groundspeed of 100 miles per hour due east and encounters a 40 mile-per-hour wind from the northwest. Both the airspeed and the compass direction must change to for the plane to maintain its groundspeed and eastward direction. Find the airspeed to maintain its groundspeed and eastward direction. Round your answer to two decimal places.

A)
B)
C)
D)
E)

Question 53

Find the coordinates of the midpoint of the line segment joining the points given below. (-6, 6, 1), (-4, 10, 11)

A) (-5, 2, 6)
B) (1, 2, 6)
C) (1, 8, 6)
D) (1, 8, 5)
E) (-5, 8, 6)

Question 54

Find a set of symmetric equations of the line through the point parallel to the vector .

A)
B)
C)
D)
E)

Question 55

A child applies the brakes on a bicycle by applying a downward force of 20 pounds on the pedal when the crank makes a angle with the horizontal (see figure). The crank is 6 inches in length. Find the torque at P. Round your answer to two decimal places. ​ ​

A) 15.32 ft-lb
B) 7.66 ft-lb
C) 91.93 ft-lb
D) 6.43 ft-lb
E) 12.86 ft-lb

Question 56

Convert the following point from cylindrical coordinates to rectangular coordinates.

A)
B)
C)
D)
E)

Question 57

Suppose a plane flies at a constant groundspeed of 300 miles per hour due east and encounters a 50 mile-per-hour wind from the northwest. Both the airspeed and the compass direction must change to for the plane to maintain its groundspeed and eastward direction. Find the airspeed to maintain its groundspeed and eastward direction. Round your answer to two decimal places. ​

A)
B)
C)
D)
E)

Question 58

Determine whether the following planes are parallel, orthogonal, or neither. If they are neither parallel nor orthogonal, find the angle of intersection.

A) The planes are neither parallel nor orthogonal, the angle of intersection is .
B) The planes are parallel.
C) The planes are neither parallel nor orthogonal, the angle of intersection is .
D) The planes are orthogonal.

Question 59

Suppose a toy wagon is pulled by exerting a force of 25 pounds on a handle that makes a angle with the horizontal as shown in the figure below. Find the work done in pulling the wagon 50 feet. Round your answer to one decimal place. ​ ​

A) 1,200.0 ft-lb
B) 427.5 ft-lb
C) 2,350.2 ft-lb
D) 1,174.6 ft-lb
E) 2,348.2 ft-lb

Question 60

Find the cross product of the unit vector .

A)
B)
C)
D)
E)

Question 61

Find the direction cosines of the vector u given below.

A)
B)
C)
D)
E)

Question 62

Given the equation of a surface of revolution , find the equation of its generating curve in the xy-plane rotating about the x-axis.

A)
B)
C)
D)
E)

Question 63

Find a set of parametric equations of the line through the point parallel to the vector .

A)
B)
C)
D)
E)

Question 64

Identify the following quadric surface.

A) elliptic paraboloid
B) elliptic cone
C) hyperboloid of two sheets
D) hyperboloid of one sheet
E) ellipsoid

Question 65

Given , find .

A) -6
B) -24
C) -5
D) 4
E) 26

Question 66

Given , find .

A)
B)
C)
D)
E)

Question 67

Given and find .

A)
B)
C)
D)
E)

Question 68

Given and find .

A)
B)
C)
D)
E)

Question 69

Find the angle between the vectors for u and v given below.

A)
B)
C)
D)
E)

Question 70

Find the distance between the point and the line given by the set of parametric equations . ​

A) /3 ​
B) 1
​
C) /3 ​
D) 3
​
E) ​/1

Question 71

Given and , find .

A)
B)
C)
D)
E)

Question 72

Find a set of parametric equations of the line through the points and .

A)
B)
C)
D)
E)

Question 73

Suppose a 48,000-pound truck is parked on a slope as shown in the figure. Assume the only force to overcome is that due to gravity. Find the force perpendicular to the hill. Round your answer to one decimal place.

A) 47,070.8 lb
B) 47,436.8 lb
C) 47,313.8 lb
D) 47,270.8 lb
E) 47,280.8 lb

Question 74

Convert the following point from rectangular coordinates to cylindrical coordinates. Give any angles in radians. ​ ​

A)
B)
C)
D)
E)

Question 75

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

A)
B)
C)
D)
E)

Question 76

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

A) ​
B) ​
C) ​
D) ​
E) ​

Question 77

Given and , find .

A)
B)
C)
D)
E)

Question 78

Both the magnitude and the direction of the force on a crankshaft change as the crankshaft rotates. Find the torque on the crankshaft using the position and data shown in the figure. Round your answer to two decimal places.

A) 2,771.28 ft-lb
B) 277.13 ft-lb
C) 226.27 ft-lb
D) 554.26 ft-lb
E) 160.00 ft-lb

Question 79

To carry a 500-pound cylindrical weight, two workers lift on the ends of short ropes tied to an eyelet on the top center of the cylinder. One rope makes a angle away from the vertical and the other makes a angle as shown in the figure below. Find the vertical component of each worker's force. Round your answer to two decimal places. ​
​ ​

A)
B)
C)
D)
E)

Question 80

Find the angle between the vectors for u and v.

A) 46.88
B) 23.20
C) 66.80
D) 42.59
E) 88.85