12.1 VECTORS Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally.

12.1

VECTORS

Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

Distance Versus DisplacementIf you start at home and walk 3 miles east and then 4 miles north, you have walked a total of 7 miles. However, your distance from home is not 7 miles, but miles, by the Pythagorean Theorem. As this example illustrates, there is a distinction between distance and displacement. Distance is a number that measures separation, while displacement consists of separation and direction. For example, walking 3 miles north and walking 3 miles east give different displacements, but both correspond to a distance of 3 miles.


● ●

●

3 miles

4 miles

Home

Destination

5 miles

22 435

Example 1A person leaves her home and walks 5 miles due east and then 3 miles northeast. How far has she walked? How far away from home is she? What is her net displacement?Solution

x2 = 52 + 32 − 2 · 5 · 3 cos 135◦ = 34 − 30(− ) = 55.213.So x = = 7.431 miles. To specify her net displacement we find the angle θ using the Law ofSines: sinθ /3 = sin 135°/7.431 or sinθ = 3( )/7.431 So sinθ = 0.285 and θ = sin-10.285 = 16.588°

The net displacement is 7.431 miles in a direction 16.588◦ north of east.

Adding Displacements Using Triangles


●

●

3 m

iles

She has walked 5+3=8 miles. To determine how far away from home she is, we use the Law of Cosines:Home

Destination

5 miles

2/2

x miles

135° 45°

213.55

θ

2/2

VectorsMany physical quantities add in the same way as displacements. Such quantities are known as vectors. Like displacements, vectors are often represented as arrows. A vector has magnitude (the arrow’s length) and direction (which way the arrow points). Examples of vectors:• Velocity: This is the speed and direction of travel• Force: This is, loosely speaking, the strength and direction of a push

or a pull.• Magnetic Fields: A vector gives the direction and intensity of a

magnetic field at a point.• Vectors in Economics: Vectors are used to keep track of prices and

quantities.• Vectors in Computer Animation: Computers generate animations by

performing enormous numbers of vector-based calculations.• Population Vectors in Biology: Populations of different animals or

age groups can be represented using vectors.Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

Vector Notation and Magnitude

We write vectors as variables with arrows over them: . The notation is intended to ensure that a vector is not mistaken for a scalar, which is another name for an ordinary number.

The length or magnitude of a vector is written || ||. Two vectors with the same magnitude are not necessarily equal, because they can have different directions.


v

v

v

v

Vector Addition and SubtractionGeometrical Addition of VectorsAs with displacements, the sum of two vectors and represented by arrows is found by joining the tail of to the head of . Then is represented by the arrow drawn from the tail of to the head of .

Geometrical Subtraction of VectorsTo subtract the vector from , we can relate it to vector addition, since .


v

v

v

v

vuw

u

u

u

u

v

vuw

u

uvwor )v(u vuw

u

v-

vuw

Scalar Multiplication

• If k > 0, then points in the same direction as and is k times as long.

• If k < 0, then points in the opposite direction as and is |k| times as long.

• If k = 0, then , the zero vector.The zero vector represents no displacement and has no direction.


vk

v

vvk

0

vk

Example 1 ExtensionWe considered a person walking 5 miles due east (vector ) and then 3 miles northeast (vector ). The vector sum was the vector from Home to Destination .

If she had gone twice as far in the due east direction, the horizontal vector would have been twice as long and we would have had the following diagram:

Vector Addition and Scalar Multiplication


●

●

3 m

iles

Using trigonometry and algebra, we determined both the magnitude and direction of

Home

Destination

5 miles

x miles

135° 45°θ u

u

v

v

vuw

vuw

vu2w

u2

v

135°

Properties of Vector Addition and Scalar Multiplication

The following properties hold true for any three vectors and any two scalars a and b:1. Commutativity of addition: 2. Associativity of addition:

3. Associativity of scalar multiplication:

4. Distributivity of scalar multiplication:

5. Identities:


vabvba

)()(

wvu

,,

vvvv

1 and 0

uvvu

)()( wvuwvu

vbuavuavbvavba

)( and )(

12.2

THE COMPONENTS OF A VECTOR


Vector ComponentsExample 1 A ship travels 200 miles in a direction that its compass says is due east. The captain then discovers that the compass is faulty and that the ship has actually been heading in a direction that is 17.4◦ to the north of due east. How far north is the ship from its intended course? SolutionIn the figure, a vector has been drawn for the ship’s actual displacement. We know that || || = 200 miles and the direction of is 17.4◦ to the north of due east. We see that can be considered the sum of two vectors, one pointing due east and the other pointing due north, so From the figure, = 200 sin 17.4◦

= 59.808 miles.


●Start

.eastnorth vvv

v

vv

v

v

northv

eastv17.4°

northv

Therefore, the ship is 59.808 miles due north of its intended course.

Unit VectorsA unit vector is a vector of length 1 unit. The unit vectors in the directions of the positive x- and y-axes are called , respectively. These two vectors are important because any vector in the plane can be expressed in terms of .


and ji

and ji

1 2 3x

1

2

3y

1

1

i

j

Vector ComponentsExample 2 Let the x-axis point east and the y-axis point north. A person walks 3 miles east and then 4 miles north. Express her displacement in terms of the unit vectors .SolutionThe unit of distance is miles, so the unit vector represents a displacement of 1 mile east, and the unit vector represents a displacement of 1 mile north. The person’s displacement can be written: Displacement = 3 miles east + 4 miles north:


● ●

●

Home

Destination

i

jiv

43

j

i

3

j

4jiv

43

and ji

Resolving a Vector in the Plane into Components

In the plane, a vector of length || || which makes an angle θ with the positive x-axis can be written in terms of its components:

where v1= || || cos θ and v2= || || sin θ


v

jvivjvivv

21sin||||cos||||

v

v

v

Vector ComponentsExample 4 In Example 1, suppose the x-axis points east and the y-axis points north. Resolve the displacement vector into components parallel to the axes.SolutionThe figure shows that We know that || || = 200 miles and the direction is 17.4◦ to the north of due east. We already computed || || = 59.808 miles. Similarly || || = 200 cos17.4° = 190.848 miles. Thus, the course taken by the ship can be written as follows:


●Start

.eastnorth vvv

v

v

v

northv

eastv17.4°

northv

To check the calculation, notice that (190.848)2 + (59.808)2 ≈ 40,000 = (200)2

eastv

.808.59848.190 jijvivv northeast

Displacement VectorsExample 6 If P is the point (−3, 1) and Q is the point (2, 4), find the components of the vector SolutionThe figure shows the vector and its components v1 and v2. The length of v1 is the horizontal distance from P to Q, so|| v1 || = 2 − (−3) = 5. The length of v2 is the vertical distance from P to Q, so || v2 || = 4 − 1 = 3. Thus, the components are v1 = 5 and v2 = 3 . We write


●P (-3,1)

PQv

2v

1v

Finding components v1 and v2 from the coordinates of the points P and Q

.35 jiPQv

.PQ

PQ

i

Q (2,4)●y

x

j

Displacement Vectors

If P = (x1, y1) and Q = (x2, y2), then the

vector from P to Q is given by


jyyixxv

)()( 1212

PQv

Vectors in n DimensionsExample 7 A balloon rises vertically a distance of 2 miles, floats west a distance of 3 miles, and floats north a distance of 4 miles. The x-axis points eastward, the y-axis points northward, and the z-axis points upward. The balloon’s displacement vector has three components: a vertical component, an eastward component, and a northward component. Therefore, this displacement is a 3-dimensional vector. We write this vector as

where is a unit vector in the positive z direction.Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

,243ntDisplaceme kji

k

12.3

APPLICATION OF VECTORS


Alternate Notation for the Components of a Vector

The vector is sometimes written • This notation can be confused with the

coordinate notation used for points. For instance, (3, 4) might mean the point x = 3, y = 4 or the vector .

• Nevertheless, this notation is useful for vectors in n dimensions.


jiv

43

ji

43

).4,3(v

Population VectorsExample 2 (a) The vectors and give the populations of the six New England states in 1995 and 2005, respectively. According to the Census Bureau, these vectors are given, in millions of people, by

= (3.28, 1.24, 6.07, 1.15, 0.99, 0.59)= (3.51, 1.32, 6.40, 1.31, 1.08, 0.62).

(a) Find . Explain its significance in terms of the population of New England.

Solution (a)We have and we subtract the entries component-wise: = (3.51–3.28, 1.32–1.24, 6.40–6.07, 1.31–1.15, 1.08–0.99, 0.62–0.59) = (0.23, 0.08, 0.33, 0.16, 0.09, 0.03).The components of give the change in population for each New England state. For instance, the population of Connecticut rose from 3.28 million in 1995 to 3.51 million in 2005, a change of 3.51 – 3.28 = 0.23 million people, so RCT = 0.23.


P

Q

P

Q

PQR

PQR

R

R

R

EconomicsExample 3 After Harry Potter, the three next most financially successful US movies in 2001 were, in order, Shrek, Pearl Harbor, and The Mummy Returns. If , , and are the revenue vectors for these three movies, then = (268,198, 436), = (199,252,301), = (202,227,192).Find , the total revenue vector for all three of these movies.Solution The total revenue, , is the sum of , , and :

= (268 + 199 + 202, 198+ 252 + 227, 436 + 301 + 192) = (669, 677, 929).


s

t

st

u

u

t

u

R

R

utsR

s

Price and Consumption VectorsA car dealership has several different models of cars in its inventory, with different prices for each model. A price vector, , gives the price of each model:

= (P1, P2, . . . , Pn).

Here P1 is the price of car model 1 and P2 is the price

of model 2, and so on. A consumption vector gives the number of each model of car purchased (or consumed) during a given month:

= (C1,C2, . . . ,Cn).


P

P

C

C

EconomicsExample 5 Let represent expenses incurred by the dealer for acquisition, insurance, and overhead, for eachmodel. What does the vector represent?Solution This represents the dealer’s profit for each model. For instance, a model that the dealer sells for $29,000 may cost $25,000 to acquire from the factory, insure, and maintain. The difference of $4,000 represents profit for the dealer. The vector keeps track of this profit for each model.


EP

EP

E

Computer Graphics: Position VectorsExample 7 A video game shows two airplanes on the screen at the points (3,5) and (7,2). Both airplanes move a distance of 3 units at an angle of 70◦ counterclockwise from the x-axis. What are the new positions of the airplanes?SolutionThe initial position of the first airplane is given by the position vector . The plane’s displacement can also be thought of as a vector. Resolving that vector, , into components:

The airplane’s final position is given by

The second airplane’s initial position is given by the position vector The second plane’s displacement, however, is exactly the same as the first airplane’s displacement . Therefore, the final position of the second airplane is given by


jipstart

53 d

.819.2026.1)70sin3()70cos3( jijid

.819.7026.4819.2026.153 jijijidpp startend

.27 jiqstart

.819.4026.8819.2026.127 jijijidqq startend

12.4

THE DOT PRODUCT


The Dot ProductExample 1A car dealer sells five different models of car. The number of each model sold each week is given by the consumption vector = (22, 14, 8, 12, 19). For instance, we see that in one week the dealer sells 22 of the first model, 14 of the second, and so forth. The price of each model is given by the vector = (19, 23, 40, 47, 32), where the units are $1000s. Thus, the price of the first model is $19,000, and so forth. Find the dealer’s weekly revenue. SolutionThe total revenue earned by the dealer each week is given by Revenue (in $1000s) = (# of cars sold · price per car) for each model:

Revenue = 22 · 19 + 14 · 23 + 8 · 40 +12 · 47 + 19 · 32 = 2232.The dealer brings in $2,232,000 each week. Since the revenue is obtained by multiplying the corresponding coordinates of and and adding, we can write revenue as a dot product:

Revenue = · .Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

P

C

C

P

C

P

We define the dot product of two vectors as follows:if = (u1, u2, . . . , un) and = (v1, v2, ... , vn) are two n-dimensional vectors, then the dot product, · , is the scalar given by · = u1v1 + u2v2 + · · · + unvn.

The Dot Product


v

v

v

u

u

u

• · = || || · || || cos θ, where θ is the angle between and .

• · = · (Commutative Law)

• · ( + ) = · + · (Distributive Law)

• · = || ||2

Properties of the Dot Product


vv

v

u

uu

u

u

u

u

vv

v

v

vv

v

u

w

w

Application of the Dot ProductExample 3(a) Find · where and(b) One person walks 3 miles east and then 4 miles north to point A. Another

person walks 2 miles east and then 5 miles north to point B. Both started from the same spot, O. What is the angle of separation of these people?

Solution(a) We have


jiv

43 vw

.52 jiw

.265423)52()43( jijiwv

(b) We see that gives the first person’s position and gives the second person’s position. The angle of separation between these two people is labeled θ in the figure. We use the formula · = || || · || || cos θ. Since || || = 5 and || || = and, · = 26, we have 26 = 5 cos θ, cos θ =26/(5 ), θ = arccos(26/(5 )) = 15.068◦.

v

w

v

v

v v

w

w

w

w

2929 29

29

θ

● B (3,4)

● A (2,5)

x

y

• Perfect alignment results in the largest possible value for · . It occurs if and are parallel, with θ = 0◦.

• Perpendicularity results in · = 0. It occurs if and are at angle of θ = 90◦.

• Perfect alignment in opposite directions results in the most negative value for · . It occurs if θ = 180◦.

What Does the Dot Product Mean?


v

v

v

u

u

u

u

u

v

v

WorkIn physics, the concept of work is represented by the dot product. Suppose you load a heavy refrigerator onto a truck. The refrigerator is on casters and glides with little effort along the floor. However, to lift the refrigerator takes a lot of work. The work done, in moving the refrigerator against the force of gravity is defined by Work = · , where is the force exerted (assumed constant) and is the displacement. If we measure distance in feet and force in pounds, work is measured in foot-pounds, where 1 foot-pound is the amount of work required to raise 1 pound a distance of 1 foot. Suppose we push the refrigerator up a ramp that makes a 10◦ angle with the floor. If the ramp is 12 ft long and the force exerted on the refrigerator is 350 lbs vertically upward, then the displacement has a magnitude of 12, and the angle θ between and is 90◦−10◦=80◦.The work done by the force is Work = || || · || || cos 80◦ = 350·12 cos 80◦ = 729.322 ft-lbs.To push the refrigerator up the ramp, we do 729.322 ft-lbs of work.


F

F

F

F

F

d

d

d

d

d

12.5

MATRICES


Table to MatrixThe table shows the latest census data (in 1000s) by age group for the six New England states

We treat the array of numbers in the table as a mathematical object in its own right, independent of the row and column headers. (continued on next slide)


CT ME MA NH RI VTunder 15 710 244 244 249 207 11915 – 24 364 145 145 142 125 6725 – 34 434 147 147 165 144 7335 – 44 572 214 214 210 166 9845 – 54 473 194 194 185 140 9455 – 64 312 118 118 115 89 5765 – 74 218 95 95 75 68 44over 74 216 83 83 60 71 35

Example of a MatrixThis rectangular grid of numbers is called a matrix, usuallywritten inside parentheses. Since the numbers in the table are populations, we use P to denote this matrix:

The individual entries are called entries in the matrix; we write pij for the entry in the ith row of the jth column. For example, p21 = 364 and p16 = 119.


3571608383216

4468759595218

5789115118118312

94140185194194473

98166210214214572

73144165147147434

67125142145145364

119207249244244710

P

If A and B are m × n matrices and k is a constant:• C = kA is an m×n matrix such that cij = kaij

This is called scalar multiplication of a matrix.

• C = A+B is an m×n matrix such that cij = aij + bij . This is called matrix addition.

• C = A−B is an m×n matrix such that cij = aij−bij. This is called matrix subtraction.

Addition, Subtraction, and Scalar Multiplication


Scalar Multiplication and Subtraction

Exercise 2 (d)Evaluate 2A – 3B given that:

Solution (d)


1273

235

068

B and

05-16

36-4

752

A

363123

12217

142820

36219

6915

01824

01032

6128

14104

1273

235

068

3

0516

364

752

232 BA

• Commutativity of addition: A+ B = B + A.

• Associativity of addition: (A + B) + C = A + (B + C)

• Associativity of scalar multiplication:k1(k2A) = (k1k2)A.

• Distributivity of scalar multiplication: (k1 + k2)A = k1A+ k2A and

k(A + B) = kA + kB.

Properties Of Scalar Multiplication and Matrix Addition


then we can write the equations

xnew = axold + byold

ynew = cxold + dyold

in the compact form

If we let

then we can write

Multiplication of a Matrix and a Vector


, ),( ),( oldoldoldnewnewnew yxPyxP

andIf

.oldnew Pdc

baP

,

dc

baA

.oldnew PP

A

Vector-Matrix EquationExample 3 Let e be the number of employed people in a certain city, and u be the number of unemployed people. We define = (e, u) as the employment vector of this city. Suppose that each year, 10% of employed peopled become unemployed, and 20% of unemployed people become employed. Find a matrix A such thatSolution We have enew = eold − 0.1 eold + 0.2 uold = 0.9 eold + 0.2 uold

unew = uold − 0.2 uold + 0.1 eold = 0.1 eold + 0.8 uold.We can rewrite this pair of equations using matrix notation:


.oldnew EE

A

E

.8.01.0

2.09.0oldoldnew EEE

A

Rotation of a Position VectorExample 5 Let = (x0, y0) be a position vector. This vector is rotated about the origin through an angle φ without changing its length. What is its new position, ? Find a matrix R such thatSolutionLet r = || || = || ||. From our definition of polar coordinates, x0 = r cos θ, y0 = r sin θ, x1 = r cos(θ +φ), and y1 = r sin(θ + φ). Notice that we can rewrite the coordinates of as follows:

x1 = r cos(θ + φ)= r(cos θ cos φ − sin θ sin φ) = (r cos θ) cos φ − (r sin θ) sin φ= x0 cos φ − y0 sin φy1 = r sin(θ + φ)= r(sin θ cos φ + cos θ sin φ) = (r sin θ) cos φ + (r cos θ) sin φ= y0 cos φ + x0 sin φ.

We can rewrite this pair of equations using a matrix R:


0P

1P

.01 PP

R

0P

1P

1P

001 cossin

sincosPPP��

R

0P

1P

Rotation of a Position VectorExample 5 continued The vector is rotated through an angle φ. Notice that this equation holds for any original vector, , and any angle of rotation,φ.


0P

001 cossin

sincosPPP��

R

x

y

● (x0, y0)

● (x1, y1)

rr

φθ

0P

12.1 VECTORS Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally.

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