Calculus Early Transcendentals

William Briggs, Lyle Cochran, Bernard Gillett

Chapter 11

Vectors and Vector-Valued Functions - all with Video Answers

Educators

Section 1

Vectors in the Plane

Problem 1

Interpret the following statement: Points have a location, but no size or direction; nonzero vectors have a size and direction, but no location.

Susanna T.

Susanna T.

Numerade Educator

Problem 2

What is a position vector?

Susanna T.

Numerade Educator

Problem 3

Draw $x$ - and $y$ -axes on a page and mark two points $P$ and $Q$. Then draw $\overrightarrow{P Q}$ and $\overrightarrow{Q P}$.

Carson Merrill

Numerade Educator

Problem 4

On the diagram of Exercise $3,$ draw the position vector that is equal to $\overrightarrow{P Q}$

Mike Gaerlan

Numerade Educator

Problem 5

Given a position vector $\mathbf{v},$ why are there infinitely many vectors equal to v?

Susanna T.

Numerade Educator

Problem 6

Explain how to add two vectors geometrically.

Mike Gaerlan

Numerade Educator

Problem 7

Explain how to find a scalar multiple of a vector geometrically.

Mike Gaerlan

Numerade Educator

Problem 8

Given two points $P$ and $Q,$ how are the components of $\overrightarrow{P Q}$ determined?

Mike Gaerlan

Numerade Educator

Problem 9

$$\text { If } \mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle \text { and } \mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle, \text { how do you find } \mathbf{u}+\mathbf{v} ?$$

Mike Gaerlan

Numerade Educator

Problem 10

If $\mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle$ and $c$ is a scalar, how do you find $c \mathbf{v} ?$

Mike Gaerlan

Numerade Educator

Problem 11

How do you compute the magnitude of $\mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle ?$

Susanna T.

Numerade Educator

Problem 12

Express the vector $\mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle$ in terms of the unit vectors $\mathbf{i}$ and $\mathbf{j}$

Mike Gaerlan

Numerade Educator

Problem 13

How do you compute $|\overrightarrow{P Q}|$ from the coordinates of the points $P$ and $Q ?$

Mike Gaerlan

Numerade Educator

Problem 14

Explain how to find two unit vectors parallel to a vector $\mathbf{v}$

Mike Gaerlan

Numerade Educator

Problem 15

How do you find a vector of length 10 in the direction of $\mathbf{v}=\langle 3,-2\rangle ?$

Mike Gaerlan

Numerade Educator

Problem 16

If a force of magnitude 100 is directed $45^{\circ}$ south of east, what are its components?

Mike Gaerlan

Numerade Educator

Problem 17

Refer to the figure and carry out the following vector operations.
Scalar multiples Which of the following vectors equals $\overrightarrow{C E} ?$ (There may be more than one correct answer.)
a. $\mathbf{v}$
b. $\frac{1}{2} \overrightarrow{H I}$
c. $\frac{1}{3} \overrightarrow{O A}$
d. $\mathbf{u}$
e. $\frac{1}{2} \vec{H}$

Carson Merrill

Numerade Educator

Problem 18

Refer to the figure and carry out the following vector operations.
Scalar multiples Which of the following vectors equals $\overrightarrow{B K}$ ? (There may be more than one correct answer.)
a. $6 \mathrm{v}$
b. $-6 v$
c. $3 \overrightarrow{H I}$
d. $3 \overrightarrow{1 H}$
e. $2 \overrightarrow{A O}$

Carson Merrill

Numerade Educator

Problem 19

Refer to the figure and carry out the following vector operations.
Scalar multiples Write the following vectors as scalar multiples of $\mathbf{u}$ or $\mathbf{v}$
a. $\overrightarrow{O A}$
b. $\overrightarrow{O D}$
c. $\overrightarrow{O H}$
d. $\overrightarrow{A G}$
e. $\overrightarrow{C E}$

Susanna T.

Numerade Educator

Problem 20

Refer to the figure and carry out the following vector operations.
Scalar multiples Write the following vectors as scalar multiples of $\mathbf{u}$ or $\mathbf{v}$
a. $\vec{H}$
b. $\vec{H})$
c. $\overrightarrow{J K}$
d. $\overrightarrow{F D}$
e. $\overrightarrow{E A}$

Carson Merrill

Numerade Educator

Problem 21

Refer to the figure and carry out the following vector operations.
Vector addition Write the following vectors as sums of scalar multiples of $\mathbf{u}$ and $\mathbf{v}$
a. $\overrightarrow{O E}$
b. $\overrightarrow{O B}$
c. $\overrightarrow{O F}$
d. $\overrightarrow{O G}$
e. $\overrightarrow{O C}$
f. $\overrightarrow{O I}$
g. $\overrightarrow{O J}$
h. $\overrightarrow{O K}$
i. $\overrightarrow{O L}$

Susanna T.

Numerade Educator

Problem 22

Refer to the figure and carry out the following vector operations.
Vector addition Write the following vectors as sums of scalar multiples of $\mathbf{u}$ and $\mathbf{v}$
a. $\overrightarrow{B F}$
b. $\overrightarrow{D E}$
c. $\overrightarrow{A F}$
d. $\overrightarrow{A D}$
e. $\overrightarrow{C D}$
f. $\overline{J D}$
g. $\overrightarrow{I I}$
h. $\overrightarrow{D B}$
i. $\overrightarrow{I L}$

Susanna T.

Numerade Educator

Problem 23

Refer to the figure and carry out the following vector operations.
Components and magnitudes Define the points $O(0,0), P(3,2)$ $Q(4,2),$ and $R(-6,-1) .$ For each vector, do the following.
(i) Sketch the vector in an $x y$ -coordinate system.
(ii) Compute the magnitude of the vector.
$$\text { a. } \overrightarrow{O P}$$
$$\text { b. } \overrightarrow{Q P}$$
$$\text { c. } \overrightarrow{R Q}$$

Carson Merrill

Numerade Educator

Problem 24

Define the points $P(-3,-1)$ $Q(-1,2), R(1,2), S(3,5), T(4,2),$ and $U(6,4)$
Sketch $\overrightarrow{P U}, \overrightarrow{T R},$ and $\overrightarrow{S Q}$ and the corresponding position vectors.

Carson Merrill

Numerade Educator

Problem 25

Define the points $P(-3,-1)$ $Q(-1,2), R(1,2), S(3,5), T(4,2),$ and $U(6,4)$
Sketch $\overrightarrow{Q U}, \overrightarrow{P T},$ and $\overrightarrow{R S}$ and the corresponding position vectors.

Carson Merrill

Numerade Educator

Problem 26

Define the points $P(-3,-1)$ $Q(-1,2), R(1,2), S(3,5), T(4,2),$ and $U(6,4)$
Find the equal vectors among $\overrightarrow{P Q}, \overrightarrow{R S},$ and $\overrightarrow{T U}$

Mike Gaerlan

Numerade Educator

Problem 27

Define the points $P(-3,-1)$ $Q(-1,2), R(1,2), S(3,5), T(4,2),$ and $U(6,4)$
Which of the vectors $\overrightarrow{Q T}$ or $\overrightarrow{S U}$ is equal to \langle 5,0\rangle $?$

Mike Gaerlan

Numerade Educator

Problem 28

Let $\mathbf{u}=\langle 4,-2\rangle, \mathbf{v}=\langle-4,6\rangle,$ and $\mathbf{w}=\langle 0,8\rangle .$ Express the following vectors in the form $\langle a, b\rangle$
$$\mathbf{u}+\mathbf{v}$$

Carson Merrill

Numerade Educator

Problem 29

Let $\mathbf{u}=\langle 4,-2\rangle, \mathbf{v}=\langle-4,6\rangle,$ and $\mathbf{w}=\langle 0,8\rangle .$ Express the following vectors in the form $\langle a, b\rangle$
$$\mathbf{w}-\mathbf{u}$$

Carson Merrill

Numerade Educator

Problem 30

Let $\mathbf{u}=\langle 4,-2\rangle, \mathbf{v}=\langle-4,6\rangle,$ and $\mathbf{w}=\langle 0,8\rangle .$ Express the following vectors in the form $\langle a, b\rangle$
$$2 \mathbf{u}+3 \mathbf{v}$$

Carson Merrill

Numerade Educator

Problem 31

Let $\mathbf{u}=\langle 4,-2\rangle, \mathbf{v}=\langle-4,6\rangle,$ and $\mathbf{w}=\langle 0,8\rangle .$ Express the following vectors in the form $\langle a, b\rangle$
$$\mathbf{w}-3 \mathbf{v}$$

Carson Merrill

Numerade Educator

Problem 32

Let $\mathbf{u}=\langle 4,-2\rangle, \mathbf{v}=\langle-4,6\rangle,$ and $\mathbf{w}=\langle 0,8\rangle .$ Express the following vectors in the form $\langle a, b\rangle$
$$10 \mathbf{u}-3 \mathbf{v}+\mathbf{w}$$

Carson Merrill

Numerade Educator

Problem 33

Let $\mathbf{u}=\langle 4,-2\rangle, \mathbf{v}=\langle-4,6\rangle,$ and $\mathbf{w}=\langle 0,8\rangle .$ Express the following vectors in the form $\langle a, b\rangle$
$$8 w+v-6 u$$

Carson Merrill

Numerade Educator

Problem 34

Let $\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,$ and $\mathbf{w}=\langle 1,0\rangle$
$$\text { Find }|\mathbf{u}+\mathbf{v}|$$

Carson Merrill

Numerade Educator

Problem 35

Let $\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,$ and $\mathbf{w}=\langle 1,0\rangle$
$$\text { Find }|-2 \mathbf{v}|$$

Carson Merrill

Numerade Educator

Problem 36

Let $\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,$ and $\mathbf{w}=\langle 1,0\rangle$
$$\text { Find }|\mathbf{u}+\mathbf{v}+\mathbf{w}|$$

Carson Merrill

Numerade Educator

Problem 37

Let $\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,$ and $\mathbf{w}=\langle 1,0\rangle$
$$\text { Find }|2 \mathbf{u}+3 \mathbf{v}-4 \mathbf{w}|$$

Carson Merrill

Numerade Educator

Problem 38

Let $\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,$ and $\mathbf{w}=\langle 1,0\rangle$
Find two vectors parallel to $\mathbf{u}$ with four times the magnitude of $\mathbf{u}$.

Mike Gaerlan

Numerade Educator

Problem 39

Let $\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,$ and $\mathbf{w}=\langle 1,0\rangle$
Find two vectors parallel to $\mathbf{v}$ with three times the magnitude of $\mathbf{v}$

Mike Gaerlan

Numerade Educator

Problem 40

Let $\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,$ and $\mathbf{w}=\langle 1,0\rangle$
Which has the greater magnitude, $2 \mathbf{u}$ or $7 \mathbf{v} ?$

Carson Merrill

Numerade Educator

Problem 41

Let $\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,$ and $\mathbf{w}=\langle 1,0\rangle$
Which has the greater magnitude, $\mathbf{u}-\mathbf{v}$ or $\mathbf{w}-\mathbf{u} ?$

Carson Merrill

Numerade Educator

Problem 42

Define the points $P(-4,1), Q(3,-4),$ and $R(2,6)$
Express $\overrightarrow{P Q}$ in the form $a \mathbf{i}+b \mathbf{j}$

Mike Gaerlan

Numerade Educator

Problem 43

Define the points $P(-4,1), Q(3,-4),$ and $R(2,6)$
Express $\overrightarrow{Q R}$ in the form $a \mathbf{i}+b \mathbf{j}$

Mike Gaerlan

Numerade Educator

Problem 44

Define the points $P(-4,1), Q(3,-4),$ and $R(2,6)$
Find the unit vector with the same direction as $\overrightarrow{Q R}$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 45

Define the points $P(-4,1), Q(3,-4),$ and $R(2,6)$
Find two unit vectors parallel to $\overrightarrow{P R}$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 46

Define the points $P(-4,1), Q(3,-4),$ and $R(2,6)$
Find two vectors parallel to $\overrightarrow{R P}$ with length 4

Christy Galilei

Christy Galilei

Numerade Educator

Problem 47

Define the points $P(-4,1), Q(3,-4),$ and $R(2,6)$
Find two vectors parallel to $\overrightarrow{Q P}$ with length 4.

Christy Galilei

Christy Galilei

Numerade Educator

Problem 48

The water in a river moves south at $10 \mathrm{mi} / \mathrm{hr} .$ A motorboat travels due east at a speed of
$20 \mathrm{mi} / \mathrm{hr}$ relative to the shore. Determine the speed and direction of the boat relative to the moving water.

Carson Merrill

Numerade Educator

Problem 49

The water in a river moves south at $5 \mathrm{km} / \mathrm{hr} .$ A motorboat travels due east at a speed of
$40 \mathrm{km} / \mathrm{hr}$ relative to the water. Determine the speed of the boat relative to the shore.

Mike Gaerlan

Numerade Educator

Problem 50

In still air, a parachute with a payload falls vertically at a terminal speed of $4 \mathrm{m} / \mathrm{s}$. Find the direction and magnitude of its terminal velocity relative to the ground if it falls in a steady wind blowing horizontally from west to east at $10 \mathrm{m} / \mathrm{s}$

Carson Merrill

Numerade Educator

Problem 51

An airplane flies horizontally from east to west at $320 \mathrm{mi} / \mathrm{hr}$ relative to the air. If it flies in a steady $40 \mathrm{mi} / \mathrm{hr}$ wind that blows horizontally toward the southwest $\left(45^{\circ}$ south of \right. west), find the speed and direction of the airplane relative to the ground.

Mike Gaerlan

Numerade Educator

Problem 52

A woman in a canoe paddles due west at $4 \mathrm{mi} / \mathrm{hr}$ relative to the water in a current that flows northwest at $2 \mathrm{mi} / \mathrm{hr} .$ Find the speed and direction of the canoe relative to the shore.

Mike Gaerlan

Numerade Educator

Problem 53

A sailboat floats in a current that flows due east at 1 $\mathrm{m} / \mathrm{s}$. Due to a wind, the boat's actual speed relative to the shore is $\sqrt{3} \mathrm{m} / \mathrm{s}$ in a direction $30^{\circ}$ north of east. Find the speed and direction of the wind.

Mike Gaerlan

Numerade Educator

Problem 54

A boat is towed with a force of 150 Ib with a rope that makes an angle of $30^{\circ}$ to the horizontal. Find the horizontal and vertical components of the force.

Mike Gaerlan

Numerade Educator

Problem 55

Suppose you pull a suitcase with a strap that makes a $60^{\circ}$ angle with the horizontal. The magnitude of the force you exert on the suitcase is 40 lb.
a. Find the horizontal and vertical components of the force.
b. Is the horizontal component of the force greater if the angle of the strap is $45^{\circ}$ instead of $60^{\circ} ?$
c. Is the vertical component of the force greater if the angle of the strap is $45^{\circ}$ instead of $60^{\circ} ?$

Carson Merrill

Numerade Educator

Problem 56

Which has a greater horizontal component, a 100-N force directed at an angle of $60^{\circ}$ above the horizontal or a 60-N force directed at an angle of $30^{\circ}$ above the horizontal?

Mike Gaerlan

Numerade Educator

Problem 57

If a $500-\mathrm{lb}$ load is suspended by two chains (see figure), what is the magnitude of the force each chain must be able to support?

Carson Merrill

Numerade Educator

Problem 58

Three forces are applied to an object, as shown in the figure. Find the magnitude and direction of the sum of the forces.

Carson Merrill

Numerade Educator

Problem 59

Determine whether the following statements are true and give an explanation or counterexample.
a. José travels from point $A$ to point $B$ in the plane by following vector $\mathbf{u},$ then vector $\mathbf{v},$ and then vector $\mathbf{w}$. If he starts at $A$ and follows $\mathbf{w},$ then $\mathbf{v},$ and then $\mathbf{u},$ he still arrives at $B$
b. Maria travels from $A$ to $B$ in the plane by following the vector $\mathbf{u} .$ By following $-\mathbf{u},$ she returns from $B$ to $A$
c. $|\mathbf{u}+\mathbf{v}| \geq|\mathbf{u}|,$ for all vectors $\mathbf{u}$ and $\mathbf{v}$
d. $|\mathbf{u}+\mathbf{v}| \geq|\mathbf{u}|+|\mathbf{v}|,$ for all vectors $\mathbf{u}$ and $\mathbf{v}$
e. Parallel vectors have the same length.
f. If $\overrightarrow{A B}=\overrightarrow{C D},$ then $A=C$ and $B=D$
g. If $\mathbf{u}$ and $\mathbf{v}$ are perpendicular, then $|\mathbf{u}+\mathbf{v}|=|\mathbf{u}|+|\mathbf{v}|$
h. If $\mathbf{u}$ and $\mathbf{v}$ are parallel and have the same direction, then $|\mathbf{u}+\mathbf{v}|=|\mathbf{u}|+|\mathbf{v}|$

Mike Gaerlan

Numerade Educator

Problem 60

Given the points $A(-2,0)$ $B(6,16), C(1,4), D(5,4), E(\sqrt{2}, \sqrt{2}),$ and $F(3 \sqrt{2},-4 \sqrt{2})$
find the position vector equal to the following vectors.
a. $A B$
b. $\overrightarrow{A C}$
c. $\overline{E F}$
d. $\overrightarrow{C D}$

Carson Merrill

Numerade Educator

Problem 61

a. Find two unit vectors parallel to $\mathbf{v}=6 \mathbf{i}-8 \mathbf{j}$
b. Find $b$ if $\mathbf{v}=\left\langle\frac{1}{3}, b\right\rangle$ is a unit vector.
c. Find all values of $a$ such that $w=a \mathbf{i}-\frac{a}{3} \mathbf{j}$ is a unit vector.

Mike Gaerlan

Numerade Educator

Problem 62

For the points $A(3,4), B(6,10), C(a+2, b+5)$ and $D(b+4, a-2),$ find the values of $a$ and $b$ such that $\overrightarrow{A B}=\overrightarrow{C D}$

Carson Merrill

Numerade Educator

Problem 63

Use the properties of vectors to solve the following equations for the unknown vector $\mathbf{x}=\langle a, b\rangle .$ Let $\mathbf{u}=\langle 2,-3\rangle$ and $\mathbf{v}=\langle 4,1\rangle$
$$10 x=u$$

Mike Gaerlan

Numerade Educator

Problem 64

Use the properties of vectors to solve the following equations for the unknown vector $\mathbf{x}=\langle a, b\rangle .$ Let $\mathbf{u}=\langle 2,-3\rangle$ and $\mathbf{v}=\langle 4,1\rangle$
$$2 x+u=v$$

Mike Gaerlan

Numerade Educator

Problem 65

Use the properties of vectors to solve the following equations for the unknown vector $\mathbf{x}=\langle a, b\rangle .$ Let $\mathbf{u}=\langle 2,-3\rangle$ and $\mathbf{v}=\langle 4,1\rangle$
$$3 x-4 u=v$$

Mike Gaerlan

Numerade Educator

Problem 66

Use the properties of vectors to solve the following equations for the unknown vector $\mathbf{x}=\langle a, b\rangle .$ Let $\mathbf{u}=\langle 2,-3\rangle$ and $\mathbf{v}=\langle 4,1\rangle$
$$-4 x=u-8 v$$

Mike Gaerlan

Numerade Educator

Problem 67

A sum of scalar multiples of two or more vectors (such as $c_{1} \mathbf{u}+c_{2} \mathbf{v}+c_{3} \mathbf{w},$ where $c_{i}$ are scalars) is called a linear combination of the vectors. Let $\mathbf{i}=\langle 1,0\rangle, \mathbf{j}=\langle 0,1\rangle$ $\mathbf{u}=\langle 1,1\rangle,$ and $\mathbf{v}=\langle-1,1\rangle$
Express \langle 4,-8\rangle as a linear combination of $i$ and $j$ (that is, find scalars $c_{1}$ and $c_{2}$ such that $\langle 4,-8\rangle=c_{1} \mathbf{i}+c_{2} \mathbf{j}$ ).

Mike Gaerlan

Numerade Educator

Problem 68

A sum of scalar multiples of two or more vectors (such as $c_{1} \mathbf{u}+c_{2} \mathbf{v}+c_{3} \mathbf{w},$ where $c_{i}$ are scalars) is called a linear combination of the vectors. Let $\mathbf{i}=\langle 1,0\rangle, \mathbf{j}=\langle 0,1\rangle$ $\mathbf{u}=\langle 1,1\rangle,$ and $\mathbf{v}=\langle-1,1\rangle$
Express \langle 4,-8\rangle as a linear combination of $\mathbf{u}$ and $\mathbf{v}$.

Mike Gaerlan

Numerade Educator

Problem 69

A sum of scalar multiples of two or more vectors (such as $c_{1} \mathbf{u}+c_{2} \mathbf{v}+c_{3} \mathbf{w},$ where $c_{i}$ are scalars) is called a linear combination of the vectors. Let $\mathbf{i}=\langle 1,0\rangle, \mathbf{j}=\langle 0,1\rangle$ $\mathbf{u}=\langle 1,1\rangle,$ and $\mathbf{v}=\langle-1,1\rangle$
For arbitrary real numbers $a$ and $b$, express $(a, b)$ as a linear combination of $\mathbf{u}$ and $\mathbf{v}$

Mike Gaerlan

Numerade Educator

Problem 70

Solve the following pairs of equations for the vectors $\mathbf{u}$ and $\mathbf{v} .$ Assume $\mathbf{i}=\langle 1,0\rangle$ and $\mathbf{j}=\langle 0,1\rangle$
$$2 \mathbf{u}=\mathbf{i}, \mathbf{u}-4 \mathbf{v}=\mathbf{j}$$

Mike Gaerlan

Numerade Educator

Problem 71

Solve the following pairs of equations for the vectors $\mathbf{u}$ and $\mathbf{v} .$ Assume $\mathbf{i}=\langle 1,0\rangle$ and $\mathbf{j}=\langle 0,1\rangle$
$$2 \mathbf{u}+3 \mathbf{v}=\mathbf{i}, \mathbf{u}-\mathbf{v}=\mathbf{j}$$

Mike Gaerlan

Numerade Educator

Problem 72

Find the following vectors.
The vector that is 3 times (3,-5) plus -9 times (6,0)

Carson Merrill

Numerade Educator

Problem 73

Find the following vectors.
The vector in the direction of \langle 5,-12\rangle with length 3

Carson Merrill

Numerade Educator

Problem 74

Find the following vectors.
The vector in the direction opposite that of (6,-8) with length 10

Carson Merrill

Numerade Educator

Problem 75

Find the following vectors.
The position vector for your final location if you start at the origin and walk along (4,-6) followed by \langle 5,9\rangle

Carson Merrill

Numerade Educator

Problem 76

An ant walks due east at a constant speed of $2 \mathrm{mi} / \mathrm{hr}$ on a sheet of paper that rests on a table. Suddenly, the sheet of paper starts moving southeast at $\sqrt{2} \mathrm{mi} / \mathrm{hr} .$ Describe the motion of the ant relative to the table.

Yingtai Xiao

Numerade Educator

Problem 77

Consider the 12 vectors that have their tails at the center of a (circular) clock and their heads at the numbers on the edge of the clock.
a. What is the sum of these 12 vectors?
b. If the 12: 00 vector is removed, what is the sum of the remaining 11 vectors?
c. By removing one or more of these 12 clock vectors, explain how to make the sum of the remaining vectors as large as possible in magnitude.
d. Consider the 11 vectors that originate at the number 12 at the top of the clock and point to the other 11 numbers. What is the sum of the vectors?

Carson Merrill

Numerade Educator

Problem 78

Three people located at $A, B$, and $C$ pull on ropes tied to a ring. Find the magnitude and direction of the force with which the person at $C$ must pull so that no one moves (the system is in equilibrium).

Mutahar Mehkri

Numerade Educator

Problem 79

Jack pulls east on a rope attached to a camel with a force of 40 ib. Jill pulls north on a rope attached to the same camel with a force of 30 Ib. What is the magnitude and direction of the force on the camel? Assume the vectors lie in a horizontal plane.

Carson Merrill

Numerade Educator

Problem 80

A 100-kg object rests on an inclined plane at an angle of $30^{\circ}$ to the floor. Find the components of the force perpendicular to and parallel to the plane. (The vertical component of the force exerted by an object of mass $m$ is its weight, which is $m g$, where $g=9.8 \mathrm{m} / \mathrm{s}^{2}$ is the acceleration due to gravity.)

Mike Gaerlan

Numerade Educator

Problem 81

Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ are vectors in the $x y$ -plane and a and $c$ are scalars.
$$\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$$

Mike Gaerlan

Numerade Educator

Problem 82

Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ are vectors in the $x y$ -plane and a and $c$ are scalars.
$$(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$$

Mike Gaerlan

Numerade Educator

Problem 83

Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ are vectors in the $x y$ -plane and a and $c$ are scalars.
$$a(c \mathbf{v})=(a c) \mathbf{v}$$

Mike Gaerlan

Numerade Educator

Problem 84

Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ are vectors in the $x y$ -plane and a and $c$ are scalars.
$$a(\mathbf{u}+\mathbf{v})=a \mathbf{u}+a \mathbf{v}$$

Mike Gaerlan

Numerade Educator

Problem 85

Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ are vectors in the $x y$ -plane and a and $c$ are scalars.
$$(a+c) \mathbf{v}=a \mathbf{v}+c \mathbf{v}$$

Mike Gaerlan

Numerade Educator

Problem 86

Use vectors to show that the midpoint of the line segment joining $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is the point $\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$ (Hint: Let $O$ be the origin and let $M$ be the midpoint of $P Q$. Draw a picture and show that
$$\left.\overrightarrow{O M}=\overrightarrow{O P}+\frac{1}{2} \overrightarrow{P Q}=\overrightarrow{O P}+\frac{1}{2}(\overrightarrow{O Q}-\overrightarrow{O P}) \cdot\right)$$

Mike Gaerlan

Numerade Educator

Problem 87

Prove that $|c \mathbf{v}|=|c||\mathbf{v}|,$ where $c$ is a scalar and $\mathbf{v}$ is a vector.

Carson Merrill

Numerade Educator

Problem 88

Assume $\overrightarrow{P Q}$ equals $\overrightarrow{R S} .$ Does it follow that $\overrightarrow{P R}$ is equal to $\overrightarrow{Q S} ?$ Explain your answer.

Carson Merrill

Numerade Educator

Problem 89

A pair of nonzero vectors in the plane is linearly dependent if one vector is a scalar multiple of the other.
Otherwise, the pair is linearly independent.
a. Which pairs of the following vectors are linearly dependent and which are linearly independent: $\mathbf{u}=\langle 2,-3\rangle$ $\mathbf{v}=\langle-12,18\rangle,$ and $\mathbf{w}=\langle 4,6\rangle ?$
b. Geometrically, what does it mean for a pair of nonzero vectors
in the plane to be linearly dependent? Linearly independent?
c. Prove that if a pair of vectors $\mathbf{u}$ and $\mathbf{v}$ is linearly independent, then given any vector $w$, there are constants $c_{1}$ and $c_{2}$ such that $\mathbf{w}=c_{1} \mathbf{u}+c_{2} \mathbf{v}$

Carson Merrill

Numerade Educator

Problem 90

Show that two nonzero vectors $\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle$ and $\mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle$ are perpendicular to each other if $u_{1} v_{1}+u_{2} v_{2}=0$

Carson Merrill

Numerade Educator

Problem 91

Let $\mathbf{u}=\langle a, 5\rangle$ and $\mathbf{v}=\langle 2,6\rangle$
a. Find the value of $a$ such that $\mathbf{u}$ is parallel to $\mathbf{v}$
b. Find the value of $a$ such that $\mathbf{u}$ is perpendicular to $\mathbf{v}$

Carson Merrill

Numerade Educator

Problem 92

Suppose $\mathbf{u}$ and $\mathbf{v}$ are vectors in the plane.
a. Use the Triangle Rule for adding vectors to explain why $|\mathbf{u}+\mathbf{v}| \leq|\mathbf{u}|+|\mathbf{v}| .$ This result is known as the Triangle Inequality.
b. Under what conditions is $|\mathbf{u}+\mathbf{v}|=|\mathbf{u}|+|\mathbf{v}| ?$

Carson Merrill

Numerade Educator