Vector-section 9.4

SECTION 9.4 Vectors 593

In simple terms, a vector (derived from the Latin vehere, meaning to carry) is a quantity that has both magnitude and direction. It is customary to represent a vector by using an arrow. The length of the arrow represents the magnitude of the vector, and the arrowhead indicates the direction of the vector.

Many quantities in physics can be represented by vectors. For example, the velocity of an aircraft can be represented by an arrow that points in the direction of movement; the length of the arrow represents speed. If the aircraft speeds up, we lengthen the arrow; if the aircraft changes direction, we introduce an arrow in the new direction. See Figure 44. Based on this representation, it is not surprising that vectors and directed line segments are somehow related.

Geometric VectorsIf P and Q are two distinct points in the xy-plane, there is exactly one line containing both P and Q [Figure 45(a)]. The points on that part of the line that joins P to Q, including P and Q, form what is called the line segment PQ [Figure 45(b)]. If we order the points so that they proceed from P to Q, we have a directed line segment from P to Q, or a geometric vector, which we denote by PQ

>. In a directed line segment PQ

>,

we call P the initial point and Q the terminal point, as indicated in Figure 45(c).

*Boldface letters are used to denote vectors, to distinguish them from numbers. For handwritten work, an arrow is placed over the letter to signify a vector. For example, we write a vector by hand as v

u.

OBJECTIVES 1 Graph Vectors (p. 595) 2 Find a Position Vector (p. 596)

3 Add and Subtract Vectors Algebraically (p. 597)

4 Find a Scalar Multiple and the Magnitude of a Vector (p. 598)

5 Find a Unit Vector (p. 598)

6 Find a Vector from Its Direction and Magnitude (p. 599)

7 Model with Vectors (p. 600)

9.4 Vectors

Figure 44

P

Q

(a) Line containing P and Q (b) Line segment PQP

QTerminalpoint

Initialpoint

(c) Directed line segment PQP

QFigure 45

The magnitude of the directed line segment PQ > is the distance from the point

P to the point Q; that is, it is the length of the line segment. The direction of PQ > is

from P to Q. If a vector v* has the same magnitude and the same direction as the directed line segmentPQ

>, we write

v = PQ >

The vector v whose magnitude is 0 is called the zero vector, 0. The zero vector is assigned no direction.

Two vectors v and w are equal, written

v = w

if they have the same magnitude and the same direction.For example, the three vectors shown in Figure 46 have the same magnitude and

the same direction, so they are equal, even though they have different initial points

R

S

P

Q

T

U

Figure 46

594 CHAPTER 9 Polar Coordinates; Vectors

and different terminal points. As a result, we find it useful to think of a vector simply as an arrow, keeping in mind that two arrows (vectors) are equal if they have the same direction and the same magnitude (length).

Adding Vectors GeometricallyThe sum v + w of two vectors is defined as follows: We position the vectors v and w so that the terminal point of v coincides with the initial point of w, as shown in Figure 47. The vector v + w is then the unique vector whose initial point coincides with the initial point of v and whose terminal point coincides with the terminal point of w.

Vector addition is commutative. That is, if v and w are any two vectors, then

v + w = w + v

Figure 48 illustrates this fact. (Observe that the commutative property is another way of saying that opposite sides of a parallelogram are equal and parallel.)

Vector addition is also associative. That is, if u, v, and w are vectors, then

u + 1v + w2 = 1u + v2 + wFigure 49 illustrates the associative property for vectors.

The zero vector 0 has the property that

v + 0 = 0 + v = v

for any vector v.

If v is a vector, then -v is the vector having the same magnitude as v, but whose direction is opposite to v, as shown in Figure 50.

Furthermore,

v + 1-v2 = 0If v and w are two vectors, we define the difference v - w as

v - w = v + 1-w2Figure 51 illustrates the relationships among v, w, v + w, and v - w.

Multiplying Vectors by Numbers GeometricallyWhen dealing with vectors, we refer to real numbers as scalars. Scalars are quantities that have only magnitude. Examples of scalar quantities from physics are temperature, speed, and time. We now define how to multiply a vector by a scalar.

If a is a scalar and v is a vector, the scalar multiple av is defined as follows:

1. If a 7 0, av is the vector whose magnitude is a times the magnitude of v and whose direction is the same as v.

2. If a 6 0, av is the vector whose magnitude is 0a 0 times the magnitude of v and whose direction is opposite that of v.

3. If a = 0 or if v = 0, then av = 0.

DEFINITION

v ! w

Initial point of v

Terminal point of w

w

v

Figure 47

Figure 48

v ! ww !

vw

v

v

w

v " v

Figure 50

v " wv !

w

w

" w

vv

Figure 51

w

u ! v

v ! w

u v

Figure 491u + v2 + w = u + 1v + w2


See Figure 52 for some illustrations.For example, if a is the acceleration of an object of mass m due to a force F

being exerted on it, then, by Newtons second law of motion, F = ma. Here, ma is the product of the scalar m and the vector a.

Scalar multiples have the following properties:

0v = 0 1v = v -1v = -v 1a + b2v = av + bv a1v + w2 = av + aw a1bv2 = 1ab2v

Graph Vectors

Graphing Vectors

Use the vectors illustrated in Figure 53 to graph each of the following vectors:

(a) v - w (b) 2v + 3w (c) 2v - w + u

Figure 54 illustrates each graph.

1

EXAMPLE 1

Solution

"1v2v

v

Figure 52

Figure 53

w uv

Now Work P R O B L E M S 9 A N D 1 1

Magnitude of VectorsWe use the symbol 7v 7 to represent the magnitude of a vector v. Since 7v 7 equals the length of a directed line segment, it follows that 7v 7 has the following properties:

Properties of 7 v 7If v is a vector and if a is a scalar, then

(a) 7v 7 0 (b) 7v 7 = 0 if and only if v = 0(c) 7 -v 7 = 7v 7 (d) 7av 7 = 0a 0 7v 7

Property (a) is a consequence of the fact that distance is a nonnegative number.Property (b) follows because the length of the directed line segment PQ

> is positive

unless P and Q are the same point, in which case the length is 0. Property (c) follows because the length of the line segment PQ equals the length of the line segment QP. Property (d) is a direct consequence of the definition of a scalar multiple.

A vector u for which 7u 7 = 1 is called a unit vector.

THEOREM

DEFINITION

Figure 54

v " w

(a) v " w

2v ! 3w

(b) 2v ! 3w

2v " w ! u

(c) 2v " w ! u

" w

v

2v3w

u

" w

2v


Find a Position VectorTo compute the magnitude and direction of a vector, we need an algebraic way of representing vectors.

An algebraic vector v is represented as

v = 8a, b9where a and b are real numbers (scalars) called the components of the vector v.

We use a rectangular coordinate system to represent algebraic vectors in the plane. If v = 8a, b9 is an algebraic vector whose initial point is at the origin, then v is called a position vector. See Figure 55. Notice that the terminal point of the position vector v = 8a, b9 is P = 1a, b2.

The next result states that any vector whose initial point is not at the origin is equal to a unique position vector.

Suppose that v is a vector with initial point P1 = 1x1 , y12, not necessarily the origin, and terminal point P2 = 1x2 , y22. If v = P1P2 >, then v is equal to the position vector

v = 8x2 - x1 , y2 - y19 (1)To see why this is true, look at Figure 56.

Figure 56v = 8a, b9 = 8x2 - x1, y2 - y19

a

bv

P # (a, b)

aA

v

P2 # (x2, y2)

P1 # (x1, y1)

x2 " x1

y2 " y1

QO

b

x

y

Triangle OPA and triangle P1P2Q are congruent. [Do you see why? The line segments have the same magnitude, so d1O, P2 = d1P1 , P22; and they have the same direction, so !POA = !P2P1Q. Since the triangles are right triangles, we have anglesideangle.] It follows that corresponding sides are equal. As a result, x2 - x1 = a and y2 - y1 = b, so v may be written as

v = 8a, b9 = 8x2 - x1 , y2 - y19Because of this result, we can replace any algebraic vector by a unique position

vector, and vice versa. This flexibility is one of the main reasons for the wide use of vectors.

Finding a Position Vector

Find the position vector of the vector v = P1P2 >, if P1 = 1-1, 22 and P2 = 14, 62.

By equation (1), the position vector equal to v is

v = 84 - 1-12, 6 - 29 = 85, 49See Figure 57.

2

DEFINITION

THEOREM

EXAMPLE 2

Solution

In WordsAn algebraic vector represents driving directions to get from the initial point to the terminal point of a vector. So, if v = 85, 49 , travel 5 units right and 4 units up from the initial point to arrive at the terminal point.

Figure 55

y

P # (a, b)

xo

v =


Two position vectors v and w are equal if and only if the terminal point of v is the same as the terminal point of w. This leads to the following result:

Equality of Vectors

Two vectors v and w are equal if and only if their corresponding components are equal. That is,

If v = 8a1 , b19 and w = 8a2 , b29 then v = w if and only if a1 = a2 and b1 = b2 .

We now present an alternative representation of a vector in the plane that is common in the physical sciences. Let i denote the unit vector whose direction is along the positive x-axis; let j denote the unit vector whose direction is along the positive y-axis. Then i = 81, 09 and j = 80, 19 , as shown in Figure 58. Any vector v = 8a, b9 can be written using the unit vectors i and j as follows:

v = 8a, b9 = a81, 09 + b80, 19 = ai + bjWe call a and b the horizontal and vertical components of v, respectively. For example, if v = 85, 49 = 5i + 4j, then 5 is the horizontal component and 4 is the vertical component.

Now Work P R O B L E M 2 9

Add and Subtract Vectors AlgebraicallyWe define the sum, difference, scalar multiple, and magnitude of algebraic vectors in terms of their components.

Let v = a1i + b1j = 8a1 , b19 and w = a2 i + b2 j = 8a2 , b29 be two vectors, and let a be a scalar. Then

v + w = 1a1 + a22i + 1b1 + b22j = 8a1 + a2 , b1 + b29 (2)v - w = 1a1 - a22i + 1b1 - b22j = 8a1 - a2 , b1 - b29 (3)

av = 1aa12i + 1ab12j = 8aa1 , ab19 (4)7v 7 = 2a21 + b21 (5)These definitions are compatible with the geometric definitions given earlier in

this section. See Figure 59.

THEOREM

3

DEFINITION

In WordsTo add two vectors, add corresponding components. To subtract two vectors, subtract corresponding components.

Figure 57

y

v #

(5, 4)

P2 # (4, 6)

P1 # ("1, 2)

O 5

5

x

Figure 58y

j

i

(0, 1)

(1, 0) x

Figure 59

a2a2

b1

b2b2

a1

y

v ! w

(a2, b2)

(a1, b1)

(a1 ! a2, b1 ! b2)

O x

(a) Illustration of property (2)$a1

$%0

b1$b1

a1

y

$v

(a1, b1)

($a1, $b1)

O x

(b) Illustration of property (4),

b1b1

y

P1 # (a1, b1)

O x

(c)

a21 ! b21

a1

Illustration of property (5):|| v || # Distance from O to P1|| v || #

w

vv

v


Adding and Subtracting Vectors

If v = 2i + 3j = 82, 39 and w = 3i - 4j = 83, -49 , find:(a) v + w (b) v - w

(a) v + w = 12i + 3j2 + 13i - 4j2 = 12 + 32i + 13 - 42j = 5i - j or

v + w = 82, 39 + 83, -49 = 82 + 3, 3 + 1-429 = 85, -19(b) v - w = 12i + 3j2 - 13i - 4j2 = 12 - 32i + 33 - 1-424j = - i + 7j or

v - w = 82, 39 - 83, -49 = 82 - 3, 3 - 1-42 9 = 8-1, 79 Find a Scalar Multiple and the Magnitude of a Vector

Finding Scalar Multiples and Magnitudes of Vectors

If v = 2i + 3j = 82, 39 and w = 3i - 4j = 83, -49 , find:(a) 3v (b) 2v - 3w (c) 7v 7(a) 3v = 312i + 3j2 = 6i + 9j or 3v = 382, 39 = 86, 99(b) 2v - 3w = 212i + 3j2 - 313i - 4j2 = 4i + 6j - 9i + 12j

= -5i + 18j or

2v - 3w = 282, 39 - 383, -49 = 84, 69 - 89, -129 = 84 - 9, 6 - 1-1229 = 8-5, 189

(c) 7v 7 = 72i + 3j 7 = 222 + 32 = 213 Now Work P R O B L E M S 3 5 A N D 4 1

For the remainder of the section, we will express a vector v in the form ai + bj.

Find a Unit VectorRecall that a unit vector u is a vector for which 7u 7 = 1. In many applications, it is useful to be able to find a unit vector u that has the same direction as a given vector v.

Unit Vector in the Direction of v

For any nonzero vector v, the vector

u = v7v 7is a unit vector that has the same direction as v.

Proof Let v = ai + bj. Then 7v 7 = 2a2 + b2 andu = v7v 7 = ai + bj2a2 + b2 = a2a2 + b2 i + b2a2 + b2 j

EXAMPLE 3

Solution

4

EXAMPLE 4

Solution

5

THEOREM


The vector u is in the same direction as v, since 7v 7 7 0. Furthermore,7u 7 = B a2a2 + b2 + b2a2 + b2 = Ba2 + b2a2 + b2 = 1

That is, u is a unit vector in the direction of v. "

As a consequence of this theorem, if u is a unit vector in the same direction as a vector v, then v may be expressed as

v = 7v 7u (6)This way of expressing a vector is useful in many applications.

Finding a Unit Vector

Find a unit vector in the same direction as v = 4i - 3j.

We find 7v 7 first. 7v 7 = 74i - 3j 7 = 216 + 9 = 5Now we multiply v by the scalar

17v 7 = 15 . A unit vector in the same direction as v isv7v 7 = 4i - 3j5 = 45 i - 35 j

Check: This vector is, in fact, a unit vector because

a 45b2 + a - 3

5b2 = 16

25+ 9

25= 25

25= 1


Find a Vector from Its Direction and MagnitudeIf a vector represents the speed and direction of an object, it is called a velocity vector. If a vector represents the direction and amount of a force acting on an object, it is called a force vector. In many applications, a vector is described in terms of its magnitude and direction, rather than in terms of its components. For example, a ball thrown with an initial speed of 25 miles per hour at an angle of 30! to the horizontal is a velocity vector.

Suppose that we are given the magnitude 7v 7 of a nonzero vector v and the direction angle a, 0! a 6 360!, between v and i. To express v in terms of 7v 7 and a, first find the unit vector u having the same direction as v.

u = v7v 7 or v = 7v 7u (7)Look at Figure 60. The coordinates of the terminal point of u are 1cos a, sin a2.

Then u = cos a i + sin a j and, from (7),

v = 7v 7 1cos a i + sin a j2 (8)where a is the direction angle between v and i.

EXAMPLE 5

Solution

6

Figure 60

1

1

y

x$

i

j (cos $, sin $)u

v


Writing a Vector When Its Magnitude and Direction Are Given

A ball is thrown with an initial speed of 25 miles per hour in a direction that makes an angle of 30! with the positive x-axis. Express the velocity vector v in terms of i and j. What is the initial speed in the horizontal direction? What is the initial speed in the vertical direction?

The magnitude of v is 7v 7 = 25 miles per hour, and the angle between the direction of v and i, the positive x-axis, is a = 30!. By equation (8),

EXAMPLE 6

Solution

Figure 61

y

x

12.5 j

21.65 i

12.5

21.65

v = 25(cos 30i + sin 30j)

25

30

Figure 62

x

$

v # 4i " 4j

"4

4

(4, "4)

Figure 63

F1 + F2

Resultant

F1

F2

Orlando

Naples Miami

Wind

N

SW E

v = 7v 7 1cos ai + sin a j2 = 251cos 30!i + sin 30!j2 = 25a232

i + 12

jb = 25232

i + 252

j

The initial speed of the ball in the horizontal direction is the horizontal component

of v, 2523

2! 21.65 miles per hour. The initial speed in the vertical direction is the

vertical component of v, 252

= 12.5 miles per hour. See Figure 61.


Finding the Direction Angle of a Vector

Find the direction angle a of v = 4i - 4j.

See Figure 62. The direction angle a of v = 4i - 4j can be found by solving

tan a = -44

= -1

Because 0! a 6 360!, the direction angle is a = 315!.


Model with VectorsBecause forces can be represented by vectors, two forces combine the way that vectors add. If F1 and F2 are two forces simultaneously acting on an object, the vector sum F1 + F2 is the resultant force. The resultant force produces the same effect on the object as that obtained when the two forces F1 and F2 act on the object. See Figure 63.

Finding the Actual Speed and Direction of an Aircraft

A Boeing 737 aircraft maintains a constant airspeed of 500 miles per hour headed due south. The jet stream is 80 miles per hour in the northeasterly direction.

(a) Express the velocity va of the 737 relative to the air and the velocity vw of the jet stream in terms of i and j.

(b) Find the velocity of the 737 relative to the ground.(c) Find the actual speed and direction of the 737 relative to the ground.

EXAMPLE 7

Solution

7

EXAMPLE 8


(a) We set up a coordinate system in which north (N) is along the positive y-axis. See Figure 64. The velocity of the 737 relative to the air is va = -500j. The velocity of the jet stream vw has magnitude 80 and direction NE (northeast), so the angle between vw and i is 45. We express vw in terms of i and j as

vw = 801cos 45! i + sin 45! j2 = 80a222 i + 222 jb = 4022 1i + j2(b) The velocity of the 737 relative to the ground vg is

vg = va + vw = -500j + 4022 1i + j2 = 4022i + 14022 - 5002j(c) The actual speed of the 737 is7vg 7 = 31402222 + 14022 - 50022 ! 447 miles per hour To find the actual direction of the 737 relative to the ground, we determine the

direction angle of vg. The direction angle is found by solving

tan a = 4022 - 5004022

Then a ! -82.7!. The 737 is traveling S7.3E.


Finding the Weight of a Piano

Two movers require a magnitude of force of 300 pounds to push a piano up a ramp inclined at an angle 20 from the horizontal. How much does the piano weigh?

Let F1 represent the force of gravity, F2 represent the force required to move the piano up the ramp, and F3 represent the force of the piano against the ramp. See Figure 65. The angle between the ground and the ramp is the same as the angle between F1 and F3 because triangles ABC and BDE are similar, so !BAC = !DBE = 20!. We wish to find the magnitude of F1. So,

sin 20! =7 F2 77F1 7 = 3007F1 7

7F1 7 = 300 lbsin 20! ! 877 lbThe piano weighs approximately 877 pounds.

An object is said to be in static equilibrium if (1) the object is at rest and (2) the sum of all forces acting on the object is zero, that is, if the resultant force is 0.

An Object in Static Equilibrium

A box of supplies that weighs 1200 pounds is suspended by two cables attached to the ceiling, as shown in Figure 66 on the next page. What are the tensions in the two cables?

We draw a force diagram using the vectors shown in Figure 67. The tensions in the cables are the magnitudes 7F1 7 and 7F2 7 of the force vectors F1 and F2 . The magnitude of the force vector F3 equals 1200 pounds, the weight of the box. Now write each force vector in terms of the unit vectors i and j. For F1 and F2 , we use equation (8). Remember that a is the angle between the vector and the positive x-axis.

Solution

EXAMPLE 9

Solution

EXAMPLE 10

Solution

Figure 64

W

N

S

vg"500

E

y

x500

vw

va # "500j

2020

A

B

D

C

E

F3

F1F2

Figure 65


F1 = 7F1 7 1cos 150!i + sin 150!j2 = 7F1 7 a - 232 i + 12 jb = - 232 7F1 7 i + 12 7F1 7 j F2 = 7F2 7 1cos 45!i + sin 45!j2 = 7F2 7 a222 i + 222 jb = 222 7F2 7 i + 222 7F2 7 j F3 = -1200j

For static equilibrium, the sum of the force vectors must equal zero.

F1 + F2 + F3 = - 232

7F1 7 i + 12 7F1 7 j + 222 7F2 7 i + 222 7F2 7 j - 1200j = 0The i component and j component each equal zero. This results in the two equations

- 232

7F1 7 + 222 7F2 7 = 0 (9)

12

7F1 7 + 222 7F2 7 - 1200 = 0 (10)We solve equation (9) for 7F2 7 and obtain 7F2 7 = 2322 7F1 7 (11)Substituting into equation (10) and solving for 7F1 7 , we obtain

12

7F1 7 + 222 a2322 7F1 7 b - 1200 = 0 12

7F1 7 + 232 7F1 7 - 1200 = 0 1 + 23

2 7F1 7 = 1200

7F1 7 = 24001 + 23 ! 878.5 pounds

Substituting this value into equation (11) yields 7F2 7 .7F2 7 = 2322 7F1 7 = 2322 # 24001 + 23 ! 1075.9 poundsThe left cable has tension of approximately 878.5 pounds, and the right cable has tension of approximately 1075.9 pounds.


Figure 67

y

30x

F1

F2

F3

45

150

Figure 66

30

30

45

45

1200pounds

The history of vectors is surprisingly complicated for such a natural concept. In the xy-plane, complex numbers do a good job of imitating vectors. About 1840, mathematicians became interested in finding a system that would do for three dimensions what the complex numbers do for two dimensions. Hermann Grassmann (18091877), in Germany, and William Rowan Hamilton (18051865), in Ireland, both attempted to find solutions.

Hamiltons system was the quaternions, which are best thought of as a real number plus a vector, and do for four dimensions what complex numbers do for two dimensions. In this system the order of multiplication matters; that is, ab 3 ba. Also, two products of

vectors emerged, the scalar (or dot) product and the vector (or cross) product.

Grassmanns abstract style, although easily read today, was almost impenetrable during the previous century, and only a few of his ideas were appreciated. Among those few were the same scalar and vector products that Hamilton had found.

About 1880, the American physicist Josiah Willard Gibbs (18391903) worked out an algebra involving only the simplest concepts: the vectors and the two products. He then added some calculus, and the resulting system was simple, flexible, and well adapted to expressing a large number of physical laws. This system remains in use essentially unchanged. Hamiltons and Grassmanns more extensive systems each gave birth to much interesting mathematics, but little of this mathematics is seen at elementary levels.

Historical Feature

Josiah Gibbs (1839 1903)


9.4 Assess Your Understanding

Concepts and Vocabulary

1. A ______ is a quantity that has both magnitude and direction.

2. If v is a vector, then v + 1-v2 = . 3. A vector u for which 7u 7 = 1 is called a(n) ______ vector. 4. If v = 6a, b7 is an algebraic vector whose initial point is

the origin, then v is called a(n) ________ vector.

5. If v = ai + bj, then a is called the _________ component of v and b is called the _________ component of v.

6. If F1 and F2 are two forces simultaneously acting on an object, the vector sum F1 + F2 is called the _________ force.

7. True or False Force is an example of a vector. 8. True or False Mass is an example of a vector.

Skill Building

In Problems 916, use the vectors in the figure at the right to graph each of the following vectors.

9. v + w 10. u + v

11. 3v 12. 4w

13. v - w 14. u - v

15. 3v + u - 2w 16. 2u - 3v + w

In Problems 1724, use the figure at the right. Determine whether the given statement is true or false.

17. A + B = F 18. K + G = F

19. C = D - E + F 20. G + H + E = D

21. E + D = G + H 22. H - C = G - F

23. A + B + K + G = 0 24. A + B + C + H + G = 0

25. If 7v 7 = 4, what is 73v 7? 26. If 7v 7 = 2, what is 7 -4v 7? In Problems 2734, the vector v has initial point P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. 27. P = 10, 02; Q = 13, 42 28. P = 10, 02; Q = 1-3, -52 29. P = 13, 22; Q = 15, 62 30. P = 1-3, 22; Q = 16, 52 31. P = 1-2, -12; Q = 16, -22 32. P = 1-1, 42; Q = 16, 22 33. P = 11, 02; Q = 10, 12 34. P = 11, 12; Q = 12, 22 In Problems 3540, find 7v 7 . 35. v = 3i - 4j 36. v = -5i + 12j 37. v = i - j 38. v = - i - j 39. v = -2i + 3j 40. v = 6i + 2j

In Problems 4146, find each quantity if v = 3i - 5j and w = -2i + 3j.

41. 2v + 3w 42. 3v - 2w 43. 7v - w 7 44. 7v + w 7 45. 7v 7 - 7w 7 46. 7v 7 + 7w 7 In Problems 4752, find the unit vector in the same direction as v.

47. v = 5i 48. v = -3j 49. v = 3i - 4j

50. v = -5i + 12j 51. v = i - j 52. v = 2i - j

v

u

w

G

B

KFAH

C

DE

53. Find a vector v whose magnitude is 4 and whose component in the i direction is twice the component in the j direction.

54. Find a vector v whose magnitude is 3 and whose component in the i direction is equal to the component in the j direction.

55. If v = 2i - j and w = xi + 3j, find all numbers x for which 7v + w 7 = 5. 56. If P = 1-3, 12 and Q = 1x, 42, find all numbers x such that the vector represented by PQ > has length 5. In Problems 5762, write the vector v in the form ai + bj, given its magnitude 7v 7 and the angle a it makes with the positive x-axis. 57. 7v 7 = 5, a = 60! 58. 7v 7 = 8, a = 45! 59. 7v 7 = 14, a = 120! 60. 7v 7 = 3, a = 240! 61. 7v 7 = 25, a = 330! 62. 7v 7 = 15, a = 315!


71. Computer Graphics The field of computer graphics utilizes vectors to compute translations of points. For example, if the point 1-3, 22 is to be translated by v = 85, 29 , then the new location will be u" = u + v = 8 -3, 29 + 85, 29 = 82, 49 . As illustrated in the figure, the point 1-3, 22 is translated to 12, 42 by v.Source: Phil Dadd. Vectors and Matrices: A Primer. www.gamedev.net/reference/articles/article1832.asp(a) Determine the new coordinates of 13, -12 if it is

translated by v = 8 -4, 59 . (b) Illustrate this translation graphically.

x"5 5

y

5

"5

(2, 4)

(5, 2)("3, 2) u'

u

v

v

72. Computer Graphics Refer to Problem 71. The points 1-3, 02, 1-1, -22, 13, 12 , and 11, 32 are the vertices of a parallelogram ABCD.(a) Find the new vertices of a parallelogram A"B"C"D" if

it is translated by v = 83, -29 .(b) Find the new vertices of a parallelogram A"B"C"D" if

it is translated by - 12

v. 73. Force Vectors A child pulls a wagon with a force of

40 pounds. The handle of the wagon makes an angle of 30! with the ground. Express the force vector F in terms of i and j.

74. Force Vectors A man pushes a wheelbarrow up an incline of 20! with a force of 100 pounds. Express the force vector F in terms of i and j.

75. Resultant Force Two forces of magnitude 40 newtons (N) and 60 N act on an object at angles of 30! and -45! with the positive x-axis, as shown in the figure. Find the direction and magnitude of the resultant force; that is, find F1 + F2 .

y

"45

30x

&&F1&& # 40 N

&&F2&& # 60 N

76. Resultant Force Two forces of magnitude 30 newtons (N) and 70 N act on an object at angles of 45! and 120! with the positive x-axis, as shown in the figure. Find the direction and magnitude of the resultant force; that is, find F1 + F2 .

y

12045

x

&&F1&& # 30 N

&&F2&& # 70 N

77. Finding the Actual Speed and Direction of an Aircraft A Boeing 747 jumbo jet maintains a constant airspeed of 550 miles per hour (mi/hr) headed due north. The jet stream is 100 mi/hr in the northeasterly direction.(a) Express the velocity va of the 747 relative to the air and

the velocity vw of the jet stream in terms of i and j.(b) Find the velocity of the 747 relative to the ground.(c) Find the actual speed and direction of the 747 relative to

the ground.

78. Finding the Actual Speed and Direction of an Aircraft An Airbus A320 jet maintains a constant airspeed of 500 mi/hr headed due west. The jet stream is 100 mi/hr in the southeasterly direction.(a) Express the velocity va of the A320 relative to the air

and the velocity vw of the jet stream in terms of i and j.(b) Find the velocity of the A320 relative to the ground.(c) Find the actual speed and direction of the A320 relative

to the ground.

79. Ground Speed and Direction of an Airplane An airplane has an airspeed of 500 kilometers per hour (km/hr) bearing N45E. The wind velocity is 60 km/hr in the direction N30W. Find the resultant vector representing the path of the plane relative to the ground. What is the ground speed of the plane? What is its direction?

80. Ground Speed and Direction of an Airplane An airplane has an airspeed of 600 km/hr bearing S30E. The wind velocity is 40 km/hr in the direction S45E. Find the resultant vector representing the path of the plane relative to the ground. What is the ground speed of the plane? What is its direction?

81. Weight of a Boat A magnitude of 700 pounds of force is required to hold a boat and its trailer in place on a ramp whose incline is 10 to the horizontal. What is the combined weight of the boat and its trailer?

82. Weight of a Car A magnitude of 1200 pounds of force is required to prevent a car from rolling down a hill whose incline is 15 to the horizontal. What is the weight of the car?

83. Correct Direction for Crossing a River A river has a constant current of 3 km/hr. At what angle to a boat dock

In Problems 6370, find the direction angle of v for each vector.

63. v = 3i + 3j 64. v = i + 23 j 65. v = -323i + 3j 66. v = -5i - 5j 67. v = 4i - 2j 68. v = 6i - 4j 69. v = - i - 5j 70. v = - i + 3j

Applications and Extensions


should a motorboat capable of maintaining a constant speed of 20 km/hr be headed in order to reach a point directly

opposite the dock? If the river is 12

kilometer wide, how long will it take to cross?

Current

Boat

Direction of boatdue to current

84. Finding the Correct Compass Heading The pilot of an aircraft wishes to head directly east but is faced with a wind speed of 40 mi/hr from the northwest. If the pilot maintains an airspeed of 250 mi/hr, what compass heading should be maintained to head directly east? What is the actual speed of the aircraft?

85. Static Equilibrium A weight of 1000 pounds is suspended from two cables as shown in the figure. What are the tensions in the two cables?

25 40

1000pounds

86. Static Equilibrium A weight of 800 pounds is suspended from two cables, as shown in the figure. What are the tensions in the two cables?

35 50

800pounds

87. Static Equilibrium A tightrope walker located at a certain point deflects the rope as indicated in the figure. If the weight of the tightrope walker is 150 pounds, how much tension is in each part of the rope?

150 pounds

3.74.2

88. Static Equilibrium Repeat Problem 87 if the angle on the left is 3.8!, the angle on the right is 2.6!, and the weight of the tightrope walker is 135 pounds.

89. Truck Pull At a county fair truck pull, two pickup trucks are attached to the back end of a monster truck as illustrated in the figure. One of the pickups pulls with a force of 2000 pounds and the other pulls with a force of 3000 pounds with an angle of 45! between them. With how much force must the monster truck pull in order to remain unmoved?

[Hint: Find the resultant force of the two trucks.]

452000

lb

3000 lb

90. Removing a Stump A farmer wishes to remove a stump from a field by pulling it out with his tractor. Having removed many stumps before, he estimates that he will need 6 tons (12,000 pounds) of force to remove the stump. However, his tractor is only capable of pulling with a force of 7000 pounds, so he asks his neighbor to help. His neighbors tractor can pull with a force of 5500 pounds. They attach the two tractors to the stump with a 40! angle between the forces as shown in the figure.(a) Assuming the farmers estimate of a needed 6-ton force

is correct, will the farmer be successful in removing the stump? Explain.

(b) Had the farmer arranged the tractors with a 25! angle between the forces, would he have been successful in removing the stump? Explain.

40550

0 lb

7000 lb

91. Static Equilibrium Show on the following graph the force needed for the object at P to be in static equilibrium.

F1

F2

F3

F4

P

Vector-section 9.4

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Transcript of Vector-section 9.4