Deck 12: Vectors and Matrices

Question 1

Is the distance between a satellite and the earth a vector or a scalar?

Answer 1

scalar

Question 2

Which of the following would be described by vectors rather than scalars?

A) The number of lions at the zoo.
B) The number of dogs and frogs at the pet store.
C) The velocity of a hot air balloon.
D) The height of your math teacher.

Answer 2

The velocity of a hot air balloon.
The height of your math teacher.

Question 3

A kite on a 26 meter string is flying with an angle of $33^{\circ}$ with the ground. What is the magnitude and the direction of the vector from the kite to the ground. Round numbers to 3 decimal places if necessary.

Answer 3

The vector points directly downward and has magnitude 14.161 meters

Question 4

Simplify $2(4 \vec{v}+6 \vec{w})+\vec{v}$

Question 5

Simplify $12 \vec{v}-3(4 \vec{v})$

Question 6

Suppose a man starts at work and drives $8 \mathrm{~km}$ due south, and then drives $13 \mathrm{~km}$ southeast. How far is the man from work? Round numbers to 3 decimal places if necessary.

Question 7

A canoe can travel $16 \mathrm{ft} \mathrm{sec}$ in still water. Suppose a river is flowing at a rate of 5 $\mathrm{ft} \mathrm{sec}$ . If the canoe is traveling downstream at an angle of $45^{\circ}$ with the current, what is the speed of the canoe? Round numbers to 3 decimal places if necessary.

Question 8

A man leaves his car and walks 2 miles northeast, 4 miles east, and then 8 miles southwest. How far is the person from his car? In what direction must he walk to head directly to his car? Round numbers to 3 decimal places if necessary.

Question 9

A man is sitting 9 meters above the ground in a tree which is 10 meters directly south of the fence corner of a field. A bird is spotted 2 meters above the ground and 8 meters directly north of the fence corner. How far is the man from the bird? Round your answer to 3 decimal places.

Question 10

A man is sitting 8 meters above the ground in a tree which is 10 meters directly south of the fence corner of a field. A bird is spotted 3 meters above the ground and 9 meters directly north of the fence corner. In what direction must the man face to look directly at the bird? Round your answer to 3 decimal places.

Question 11

A horse runs at a constant speed of $18 \mathrm{~m} \mathrm{sec}$ . He starts at a fence and his path makes an angle of $13^{\circ}$ with the fence. After 9 seconds, how far is he from the fence? Round numbers to 3 decimal places if necessary.

Question 12

What properties of vector addition and scalar multiplication are necessary to show $(5+a)(\vec{v}+\vec{w})=5 \vec{w}+a \vec{w}+5 \vec{v}+a \vec{v}$

Question 13

Does $-3 \vec{v}+5 \vec{v}$ point in the same direction as $\vec{v}$ , the opposite direction, or neither?

Question 14

Let $\vec{v}$ be a vector of length 3 pointing $30^{\circ}$ north of east. Find the length and direction of $23 \vec{v}$ .

Question 15

Let $\vec{v}$ be a vector of length 3 pointing $30^{\circ}$ north of east. Find the length and direction of $-4 \vec{v}$ .

Question 16

Jack and Jill begin walking away from a water well. Jill walks 2 meters north, and then 12 meters east. When Jill stops, Jack is twice as far from the well as she is. How far is Jack from the well? Round numbers to 3 decimal places if necessary.

Question 17

Jack and Jill begin walking away from a water well. Jill walks 3 meters north, and then 14 meters northeast. When Jill stops, Jack is half as far from the well as she is. How far is Jack from the well? Round numbers to 3 decimal places if necessary.

Question 18

Jack and Jill begin walking away from a water well. Jill walks 4 meters west, and then 2 meters north. When Jill stops, Jack is three times as far from the well as she is, but in the opposite direction. If $\vec{v}$ is the vector pointing from Jack to the well, find the length and direction of $\vec{v}$ . Round numbers to 3 decimal places if necessary.

Question 19

Simplify the following: $3(2 \vec{v}+7 \vec{w})-(-3 \vec{v}-7 \vec{w})$

Question 20

Simplify the following: $2(6 \vec{v}-\vec{w}-\vec{p})+9(\vec{v}-4 \vec{w}+\vec{p})$

Question 21

The figure below shows the vector $\vec{v}=$ ---------- $\vec{i}+$ ---------- $\vec{j}$ .

Question 22

The vector starting at the point $P=(2,4)$ and ending at the point $Q=(7,5)$ can be resolved into the components ----------------- $\vec{i}+$ -------------- $\vec{j}$

Question 23

Find the length of the vector $\vec{v}=-2.5 \vec{i}+3.5 \vec{j}-4.1 \vec{k}$ to 3 decimal places.

Question 24

A boat in the ocean at point $(39,58,0)$ is looking at the light on a lighthouse at point $(39,0,25)$ . 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 ------------- $\vec{i}+$ ---------- $\vec{j}+$ ---------- $\vec{k}$ .

Question 25

A car travels 19 miles south and then 23 miles east. Which of the following is true?

A) Its net displacement is 29.83 miles.
B) Its net displacement is 29.83 miles $39.56^{\circ}$ south of east.
C) Its net displacement is 42 miles.
D) Its net displacement is 42 miles $39.56^{\circ}$ south of east.

Question 26

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 $2 \vec{i}-6 \vec{j}+0.5 \vec{k}$ represent?

A) A displacement of $2 \mathrm{~km}$ east, followed by a displacement of $6 \mathrm{~km}$ north and a displacement of $0.5 \mathrm{~km}$ up.
B) A displacement of $2 \mathrm{~km}$ west, followed by a displacement of $6 \mathrm{~km}$ north and a displacement of $0.5 \mathrm{~km}$ up.
C) A displacement of $2 \mathrm{~km}$ east, followed by a displacement of $6 \mathrm{~km}$ south and a displacement of $0.5 \mathrm{~km}$ up.
D) A displacement of $2 \mathrm{~km}$ west, followed by a displacement of $6 \mathrm{~km}$ south and a displacement of $0.5 \mathrm{~km}$ up.

Question 27

For what value of $x$ do $6 \vec{i}-x \vec{j}+(x-2) \vec{k}$ and $x \vec{i}-(x+1) \vec{j}+3 \vec{k}$ have the same length?

Question 28

If $\|\vec{v}\|=1.3$ , what is $\|-\vec{v}\|$ ?

Question 29

Find the length of the vector $-2 \vec{v}+4 \vec{w}$ if $\vec{v}=\vec{i}+2 \vec{j}-3 \vec{k}$ and $\vec{w}=-2 \vec{j}$ .

Question 30

Let $P=(2,3)$ and $Q=(-3,-1)$ . Write the displacement vector $\overrightarrow{Q P}$ in component form.

Question 31

Let $P=(6,2,1)$ and $Q=(-2,0,5)$ . Write the displacement vector $\overrightarrow{P Q}$ in component form.

Question 32

If $\vec{v}=-2 \vec{i}+2 \vec{j}-3 \vec{k}$ . Find $c$ such that $\vec{w}=-6 \vec{i}+6 \vec{j}+c \vec{k}$ is parallel to $\vec{v}$ .

Question 33

Let $P=(-1,3)$ and $Q=(2,4)$ . Find a vector of length 9 pointing in the opposite direction of $\overrightarrow{P Q}$ .

Question 34

A cannon is fired at a $15^{\circ}$ angle with the ground at a speed of $81 \mathrm{ft} / \mathrm{sec}$ . Resolve the velocity vector into horizontal and vertical components.

Question 35

The vectors $\vec{u}=8 \vec{i}-9 \vec{j}+4 \vec{k}$ and $\vec{v}=-16 \vec{i}+18 \vec{j}+8 \vec{k}$ are parallel.

Question 36

The vectors $\vec{u}=4 \vec{i}+5 \vec{j}-6 \vec{k}$ and $\vec{v}=-12 \vec{i}-15 \vec{j}+18 \vec{k}$ are parallel.

Question 37

In the figure below, each square is 6 units along each side. In component form, $\vec{u}=$ ------------- $\vec{i}+$ ---------- $\vec{j}$ .

Question 38

In the figure below, each square is 9 units along each side. In component form,

Question 39

In the figure below, each square is 7 units along each side. The vector perpendicular to the displacement vector $\overrightarrow{R S}$ is ----------- $\vec{i}+\ldots \vec{j}$ .

Question 40

$(\vec{i}-\vec{j})-2(5 \vec{i}-\vec{j})=-----------\vec{i}+\ldots \vec{j}$

Question 41

Perform the computation $3(3 \vec{i}-\vec{j})+5 \vec{j}$ .

Question 42

Perform the computation $(5 \vec{i}-2 \vec{j})+(5 \vec{i}+4 \vec{j})$ .

Question 43

A particle in equilibrium is acted upon by three forces, two of which have components $4 \vec{i}+6 \vec{j}-9 \vec{k}$ and $9 \vec{i}-4 \vec{j}+3 \vec{k}$ . The components of the third must be ------ $\underline{\vec{i}}+\ldots \vec{j}+\ldots \vec{k}$ .

Question 44

As $t$ changes, what happens to the tip of the vector $\vec{r}=(1+4 t) \vec{i}+(2-3 t) \vec{j}$ ?

A) It moves in a straight line parallel to the vector $3 \vec{i}+1 \vec{j}$ .
B) It moves in a straight line parallel to the vector $\vec{i}+2 \vec{j}$ .
C) It moves in a straight line parallel to the vector $4 \vec{i}-3 \vec{j}$ .
D) None of the above.

Question 45

The unit vector from the point $(4,3)$ toward the point $(5,5)$ has its head at the point ( -------------,------------). Round to 2 decimal places.

Question 46

Let $\vec{P}=\left(p_{1}, p_{2}, p_{3}\right)$ give the profit a salesman makes from selling three different models of computers. Let $\vec{S}=\left(s_{1}, s_{2}, s_{3}\right)$ give the number of each computer model sold each week. What does $\vec{P} \cdot \vec{S}$ represent?

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

Question 47

A retailer's total monthly sales of three different models of television is given by the vector $\vec{S}=(14,27,20)$ . If the sales for each model go up by 7 the next month, what is $\vec{Q}$ , the next month's total sales?

Question 48

A retailer's total monthly sales of three different models of television is given by the vector $\vec{S}=(12,25,12)$ . If the sales for each model go down by $15 \%$ the next month, what is $\vec{Q}$ , the next month's total sales? Round entries to the nearest whole number.

Question 49

If $R=(2,3,4,5,6)$ and $S=(0,1,4,5,9)$ , then what is $\vec{A}=4 \vec{R}-3 \vec{S}$ ?

Question 50

A snow cone stand sells three sizes of snow cones: small, medium, and large. Let $\vec{N}=(56,140,83)$ give the number of each type of cone sold in one day. Let $\vec{P}=(1.50,2.00,2.50)$ give the price (in dollars) charged for each size of snow cone, $\vec{C}=(0.30,0.40,0.50)$ give the cost (also in dollars) of making each size of snow cone, and $M=(150,200,200)$ 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 51

A snow cone stand sells three sizes of snow cones: small, medium, and large. Let $\vec{N}=(80,95,97)$ give the number of each type of cone sold in one day. Let $\vec{P}=(1.50,2.00,2.50)$ give the price (in dollars) charged for each size of snow cone, $\vec{C}=(0.30,0.40,0.50)$ give the cost (also in dollars) of making each size of snow cone, and $M=(150,200,200)$ 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 medium snow cones that day?

Question 52

If $R=(2,3,4,5,6)$ and $S=(0,1,3,4,7)$ , then what is $\vec{p}=\frac{\vec{R}}{2}+\frac{4 \vec{S}}{4}$ ?

Question 53

If $\vec{\alpha}=(1.1,3.7,4.1,5.6)$ and $\beta=(10.5,10.7,10.8,11.1)$ , find $\vec{\gamma}=6.5(\vec{\alpha}-\vec{\beta})$ .

Question 54

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 $\vec{r}=(71,49,87,98,81,63)$ . Their scores (out of 100) on the oral section are given by the vector $\vec{s}=(79,74,82,95,97,66)$ . Find the vector giving their composite score if the written part counts twice as much as the oral part.

Question 55

If $\vec{A}=(1,2,-1,3)$ and $\vec{B}=(4,8,2,0)$ , find $\vec{C}=3 \vec{A}-5 \vec{B}$

Question 56

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

Question 57

The rectangle with vertices $(1,0),(1,4),(2,0)$ , and $(2,4)$ is rotated through an angle of $21^{\circ}$ about the origin. What are the coordinates of the new rectangle? Round numbers to 3 decimal places if necessary.

Question 58

Let the student vector $\vec{P}=(121,129,119,119)$ 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 $\vec{R}$ ? Round to the nearest whole number.

Question 59

Let $\vec{A}=(7,4,7,2)$ and $\vec{B}=(4,9,3,5)$ .
Find $\frac{\vec{A}}{2}+\vec{B}$

Question 60

Let $\vec{A}=(3,2,7,2)$ and $\vec{B}=(6,8,3,5)$ .
Find $\vec{A}-3 \vec{B}$ .

Question 61

Let the student vector $\vec{P}=(120,126,117,121)$ 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 $\vec{R}$ ?

Question 62

Let the student vector $\vec{P}=(123,129,119,118)$ 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 $\vec{R}$ ?

Question 63

Given the vector $\vec{N}=\{1,5,9\}$ find vector $\vec{B}=2 \vec{N}$ .

Question 64

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 $60^{\circ}$ 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 65

A cyclist goes at 6 mph due north and feels the wind coming against him at a relative velocity of 3 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 66

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 67

For $\vec{q}=3 \vec{i}+\vec{j}-5 \vec{k}$ and $\vec{r}=3 \vec{i}-5 \vec{j}+\vec{k}$ , what is $\vec{q} \cdot \vec{r}$ ?

Question 68

For $\vec{q}=3 \vec{i}+\vec{j}-4 \vec{k}$ and $\vec{r}=-2 \vec{i}-2 \vec{j}+\vec{k}$ , $(\vec{r} \cdot \vec{r}) \vec{q}=\ldots \vec{i}+\ldots \vec{j}+\ldots \vec{k}$ .

Question 69

For $\vec{q}=-3 \vec{i}+\vec{j}-3 \vec{k}$ and $\vec{r}=-2 \vec{i}-3 \vec{j}+\vec{k}$ , what is $((\vec{r} \cdot \vec{q}) \vec{r}) \cdot \vec{q}$ ?

Question 70

A model pyramid is built using four equilateral triangles connected to a square base. If the length of one side of the base is 11 inches, how many inches high is the pyramid? Round to 2 decimal places.

Question 71

A vector of length 7 that points in the same direction as $2 \vec{i}+\vec{j}-6 \vec{k}$ is $\vec{i}+\ldots \vec{j}+\ldots \vec{k}$ . Give each answer to 3 decimal places.

Question 72

For what value of $x$ are $16 \vec{i}-12 \vec{j}+4 \vec{k}$ and $x \vec{i}-(x-1) \vec{j}+\vec{k}$ parallel?

Question 73

In the figure below, the angle between the $x$ -axis and the vector $\overrightarrow{R S}$ is------------° Round to the nearest whole number.

Question 74

An airplane is flying at an airspeed of 620 km/hr in a crosswind blowing from the southeast at a speed of 55 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 75

Let the vector $\vec{t}=\left(x_{1}, y_{1}, x_{2}, y_{2}, x_{3}, y_{3}\right)$ describe a triangle with vertices at $\left(x_{1}, y_{1}\right)$ , $\left(x_{2}, y_{2}\right)$ , and $\left(x_{3}, y_{3}\right)$ . If the triangle $\vec{u}=(4,2,6,2,4,4)$ is rotated through an angle of $30^{\circ}$ clockwise about the origin, what is the resulting vector? Round each entry to 2 decimal places.

Question 76

Three people stand in the middle of a field. The first person walks 15 yards north and then 15 yards east. The second person remains where he is. The third person walks 20 yards north and then 9 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 77

How many ft-lbs of work are required to lift a 371 pound refrigerator 3 inches straight up? Round to 2 decimal places.

Question 78

The angle between the vectors $5 \vec{i}-5 \vec{j}+\vec{k}$ and $4 \vec{i}+4 \vec{j}-\vec{k}$ is---------. Round to the nearest whole number.

Question 79

Do the points $(11,19),(13,15),(17,17)$ , and $(15,21)$ form a square?

Question 80

An airplane is flying at an airspeed of 600 km/hr in a crosswind blowing from the southeast at a speed of 50 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.