Deck 12: Vectors and Matrices

Question 1

A particle in equilibrium is acted upon by three forces, two of which have components and . The components of the third must be .

Answer 1

a)-12
b)-1
c)4

Question 2

As t changes, what happens to the tip of the vector ?

A)
B)
C)
D) None of the above.

Answer 2

Question 3

The unit vector from the point (1, 3) toward the point (3, 5) has its head at the point ( _____, _____ ). Round to 2 decimal places.

Answer 3

a) 1.71
b)3.71

Question 4

Let Give the profit a salesman makes from selling three different models of computers. Let give the number of each computer model sold each month. What does represent?

A) A scalar giving the total profit for one month from selling all three models of computers.
B) A scalar giving the average profit for one month from selling each computer model.
C) A vector giving the profit for one month for the three models of computers.
D) A vector giving the average profit for one month for the three models of computers.

Question 5

A retailer's total monthly sales of three different models of television is given by the vector . If the sales for each model go up by 8 the next month, what is , the next month's total sales?

Question 6

A retailer's total monthly sales of three different models of television is given by the vector https://d2lvgg3v3hfg70.cloudfront.net/TB10073/. If the sales for each model go down by 5% the next month, what is , the next month's total sales? Round entries to the nearest whole number.

Question 7

If and , then what is ?

Question 8

A snow cone stand sells three sizes of snow cones: small, medium, and large. Let give the number of each type of cone sold in one day. Let give the price (in dollars) charged for each size of snow cone, give the cost (also in dollars) of making each size of snow cone, and give the maximum number of each size that can be sold (because of the number of each size cup on hand.) What is the total number of cups left in the stand at the end of the day?

Question 9

A snow cone stand sells three sizes of snow cones: small, medium, and large. Let give the number of each type of cone sold in one day. Let give the price (in dollars) charged for each size of snow cone, give the cost (also in dollars) of making each size of snow cone, and give the maximum number of each size that can be sold (because of the number of each size cup on hand). How much money was spent making the small snow cones that day?

Question 10

If and , then what is ?

Question 11

If and , find .

Question 12

There are six people taking a vocational exam with both oral and written parts. Their scores (out of 100) on the written section are given by the vector . Their scores (out of 100) on the oral section are given by the vector . Find the vector giving their composite score if the written part counts twice as much as the oral part.

Question 13

If and , find

Question 14

An airplane flies at an airspeed of 500 km/hr in a cross-wind that is blowing from the southwest at a speed of 29 km/hr. What direction should the plane fly to end up going due south?

Question 15

The rectangle with vertices (1,0), (1,4), (2,0), and (2,4) is rotated through an angle of about the origin. What are the coordinates of the new rectangle?

Question 16

Let the student vector be the number of ninth, tenth, eleventh, and twelfth graders in a high school. If the number of students in each grade increases by 5%, what is the new student vector ? Round to the nearest whole number.

Question 17

Let and .Find .

Question 18

Let and .Find .

Question 19

Let the student vector be the number of ninth, tenth, eleventh, and twelfth graders in a high school. If the number of students in each grade increases by 5, what is the new student vector ?

Question 20

Let the student vector be the number of ninth, tenth, eleventh, and twelfth graders in a high school. If the number of students in each grade doubles, what is the new student vector ?

Question 21

Given that and , let , where . What is ?

Question 22

Given that and , let , where . What is ?

Question 23

Given and , does ?

Question 24

Given , , and , what is ?

Question 25

Given , , and what is ?

Question 26

Given , , , and , does ?

Question 27

A snow cone stand sells three sizes of snow cones: small, medium, and large. Let give the number of each type of cone sold in one day. Let give the price charged for each size of snow cone, give the cost of making each size of snow cone, andGive the maximum number of each size that can be sold (because of the number of each size cup on hand.) Let . Which of the following gives the maximum amount of money that can be received from one day's sales?

A)
B)
C)
D)

Question 28

A country has two main political parties. The vector Gives the number of people who are members of the first party, the second party, and neither party, respectively. Suppose

. Which party's members are the most loyal?

A) The second party
B) The first party

Question 29

A country has two main political parties. The vector gives the number of people who are members of the first party, the second party, and neither party, respectively. Suppose . What percent of the members of the first party switch to the second party each year?

Question 30

A country has two main political parties. The vector gives the number of people who are members of the first party, the second party, and neither party, respectively. Suppose . Let be the population vector (in tens of thousands) in the year 2005. What is ?

Question 31

A country has two main political parties. The vector gives the number of people who are members of the first party, the second party, and neither party, respectively. Suppose . Let be the population vector (in tens of thousands) in the year 2005. What is ?

Question 32

The inverse of a matrix , denoted by , is such that if , then . For a matrix , the inverse is given by , where . is undefined if D = 0. Let . Does ?

Question 33

Let , , and . What is ?

Question 34

In certain cases, there is a nonzero vector and a scalar for a matrix such that . The vector is called an eigenvector of with eigenvalue . Let with eigenvector . What is its eigenvalue?

Question 35

Let be the population vector of a city giving o, the number of people who own their home and r, the number of people who rent. Suppose that each year, 3% of the people who own their home move to a rental and 11% of the people who rent purchase a home and become owners. Let be such that . What is ?

Question 36

Let and .

A) Find
B)
C)

Question 37

A) A+B
B)
C)
D)
E)
F)

Question 38

Question 39

Question 40

Question 41

Let and . Find such that .

Question 42

Let . Find

Question 43

A particle is acted on by two forces, one of them to the west and of magnitude 0.5 dynes (a dyne is a unit of force), and the other in the direction 60 north of east and of magnitude 1 dyne. A third force acting upon the particle that would keep it at equilibrium has a magnitude of ______ dynes and points _____ (north / south / east / west). Round the first answer to 2 decimal places.

Question 44

A cyclist goes at 7 mph due north and feels the wind coming against him at a relative velocity of 5 mph due west. The actual velocity of the wind is _____ mph at an angle of ______ west of north. Round both answers to 1 decimal place.

Question 45

A gun mounted on a train points vertically upward. The train moves horizontally due east at 80 mph and the gun fires a bullet with a muzzle velocity of 80 mph. The speed and direction of the bullet relative to the ground is _______ mph toward the east at ______ to the horizontal. Round the first answer to 2 decimal places.

Question 46

For and , what is ?

Question 47

For and , .

Question 48

For and , what is ?

Question 49

A model pyramid is built using four equilateral triangles connected to a square base. If the length of one side of the base is 19 inches, how many inches high is the pyramid? Round to 2 decimal places.

Question 50

A vector of length 8 that points in the same direction as is . Give each answer to 3 decimal places.

Question 51

For what value of x are and parallel?

Question 52

In the figure below, the angle between the x-axis and the vector is ______ . Round to the nearest whole number.

Question 53

An airplane is flying at an airspeed of 660 km/hr in a crosswind blowing from the southeast at a speed of 60 km/hr. To end up going due east, the plane should head _____ south of east and will have a speed of ______km/hr relative to the ground. Round each answer to 2 decimal places.

Question 54

Let the vector describe a triangle with vertices at , , and . If the triangle is rotated through an angle of 28 clockwise about the origin, what is the resulting vector? Round each entry to 2 decimal places.

Question 55

Three people stand in the middle of a field. The first person walks 11 yards north and then 14 yards east. The second person remains where he is. The third person walks 18 yards north and then 8 yards east. The angle formed by drawing a line from the first person to the second person to the third person is _____ . Round to the nearest whole number.

Question 56

How many ft-lbs of work are required to lift a 324 pound refrigerator 8 inches straight up? Round to 2 decimal places.

Question 57

The angle between the vectors and is _____
. Round to the nearest whole number.

Question 58

Do the points (11, 8), (12, 6), (13, 7), and (13, 8) form a square?

Question 59

An airplane is flying at an airspeed of 690 km/hr in a crosswind blowing from the southeast at a speed of 45 km/hr. To end up going due west, the plane should head _____ south of west and will have a speed of ______km/hr relative to the ground. Round each answer to 2 decimal places.

Question 60

For what value of x are and perpendicular?

Question 61

How many ft-lbs of work are done pushing a 15-lb stroller with a 25 lb toddler in it up a 60 yard hill if the slope of the hill is 20 ? Round to 1 decimal place.

Question 62

Which of the following vectors are parallel to ? Mark all that apply.

A)
B)
C)
D)
E)

Question 63

The figure below shows the vector

.

Question 64

The vector starting at the point P = (0, 2) and ending at the point Q = (1, 4) can be resolved into the components

Question 65

Find the length of the vector to 3 decimal places.

Question 66

A boat in the ocean at point (37, 58, 0) is looking at the light on a lighthouse at point (37, 0, 21). The light house is 3 units high, and sits at the edge of a cliff above the ocean. The displacement vector between the boat and the base of the lighthouse is
.

Question 67

A car travels 19 miles south and then 24 miles east. Which of the following is true?

A)
B)
C)
D)

Question 68

Let the x-axis point east, the y-axis point north, and the z-axis point upward, and let the unit vectors be in kilometers. What does the vector represent?

A) A displacement of 5 km east, followed by a displacement of 2 km north and a displacement of 2 km up.
B) A displacement of 5 km west, followed by a displacement of 2 km north and a displacement of 2 km up.
C) A displacement of 5 km east, followed by a displacement of 2 km south and a displacement of 2 km up.
D) A displacement of 5 km west, followed by a displacement of 2 km south and a displacement of 2 km up.

Question 69

For what value of x do and have the same length?

Question 70

If , what is ?

Question 71

Find the length of the vector if and
.

Question 72

Let and . Write the displacement vector in component form.

Question 73

Let and . Write the displacement vector in component form.

Question 74

If . Find such that is parallel to .

Question 75

Let and . Find a vector of length 6 pointing in the opposite direction of .

Question 76

A cannon is fired at a angle with the ground at a speed of 80 ft/sec. Resolve the velocity vector into horizontal and vertical components.

Question 77

The vectors and are parallel.

Question 78

The vectors and are parallel.

Question 79

In the figure below, each square is 4 units along each side. In component form,

.

Question 80

In the figure below, each square is 1 units along each side. In component form, .