1 2011 Pearson Education, Inc. All rights reserved 1 2010 Pearson Education, Inc. All rights reserved 2011 Pearson Education, Inc. All rights reserved.

1© 2011 Pearson Education, Inc. All rights reserved 1© 2010 Pearson Education, Inc. All rights reserved

© 2011 Pearson Education, Inc. All rights reserved

Chapter 7

Applications of Trigonometric

Functions

OBJECTIVES

© 2011 Pearson Education, Inc. All rights reserved 2

VectorsSECTION 7.5

1

2

Represent vectors geometrically.Represent vectors algebraically.Find a unit vector in the direction of v.Write a vector in terms of its magnitude and direction.Use vectors in applications.

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5

3© 2011 Pearson Education, Inc. All rights reserved

Many physical quantities such as length, area, volume, mass, and temperature are completely described by their magnitudes in appropriate units. Such quantities are called scalar quantities.

Other physical quantities such as velocity, acceleration, and force are completely described only if both a magnitude (size) and a direction are specified. Such quantities are called vector quantities.

VECTORS


GEOMETRIC VECTORS

A vector can be represented geometrically by a directed line segment with an arrow-head. The arrow specifies the direction of the vector, and its length describes the magnitude.

The tail of the arrow is the vector’s initial point, and the tip of the arrow is its terminal point.

Vectors are denoted by lowercase boldfaced type.

With vectors, real number are scalars. Scalars are denoted by lowercase italic type.


If the initial point of a vector v is P and the terminal point is Q, we write .PQv

The magnitude (or norm)of a vector ,PQv

denoted by , or ,PQv

is the length of the vector v and is a scalar quantity.

GEOMETRIC VECTORS


EQUIVALENT VECTORS

Two vectors having the same length and same direction are called equivalent vectors.


EQUIVALENT VECTORS

Equivalent vectors are regarded as equal even though they may be located in different positions.


ZERO VECTOR

The vector of length zero is called the zero vector and is denoted by 0. The zero vector has zero magnitude and arbitrary direction.

If vectors v and a, as in thefigure to the right, have thesame length and oppositedirection, then a is theopposite vector of v andwe write a = –v.


Let v and w be any two vectors. Place the vector w so that its initial point coincides with the terminal point of v.

The sum v + w is the resultant vector whose initial point coincides with the initial point of v, and whose terminal point coincides with the terminal point of w. v

wv + w

GEOMETRIC VECTOR ADDITION


w

v + w

w

v

v

w + v

As shown in the figure, v + w = w + v.

The sum coincides with the diagonal of the parallelogram determined by v and w when v and w have the same initial point.

GEOMETRIC VECTOR ADDITION


VECTOR SUBTRACTION

For any two vectors v and w, v – w = v + (–w), where –w is the opposite of w.

w

v – wv

–w

–w

w

v – wv


SCALAR MULTIPLES OF VECTORS

Let v be a vector and c a scalar (a real number). The vector cv is called the scalar multiple of v.

If c > 0, cv has the same direction as v and magnitude c||v||.

If c < 0, cv has the opposite direction as v and magnitude |c| ||v||.

If c = 0, cv = 0v = 0.


EXAMPLE 1 Geometric Vectors

Use the vectors u, v, and w in the figure to the right to graph each vector.a. u – 2w b. 2v – u + w

Solutiona. u – 2w


EXAMPLE 1 Geometric Vectors

Solution continuedb. 2v – u + w


ALGEBRAIC VECTORS

Specifying the terminal point of the vector will completely determine the vector. For the position vector v with initial point at the origin O and terminal point at P(v1, v2), we denote the vector by

A vector drawn with its initial point at the origin is called a position vector.

1 2, .OP v v v


22

21 vv v

The magnitude of position vector follows directly from the Pythagorean Theorem.

Notice the difference between the notations for the point (v1, v2) and the position vector

We call v1 and v2 the components of the vector v; v1 is the first component, and v2 is the second component.

1 2, .v v

1 2,v vv

ALGEBRAIC VECTORS


ALGEBRAIC VECTORSIf equivalent vectors, v and w, are located so that their initial points are at the origin, then their terminal points must coincide.

Thus, for the vectors

v = w if and only ifv1 = w1 and v2 = w2.

,, and , 2121 wwvv wv

R


REPRESENTING A VECTORAS A POSITION VECTOR

The vector with initial point P(x1, y1) and terminal

point Q(x2, y2) is equal to

the position vectorw x2 x1, y2 y1 .

PQ

R


EXAMPLE 2 Representing a Vector in the Cartesian Plane

Let v be the vector with initial point P(4, –2) and terminal point Q(–1, 3). Write v as a position vector.Solutionv has• initial point P(4, –2), so x1 = 4 and y1 = –2.• terminal point Q(–1, 3), so x2 = –1 and y2 = 3.

1 4,3 2 v2 1 2 1,x yx y vSo,

5,5v


ARITHMETIC OPERATIONS AND PROPERTIES OF VECTORS

If are vectors and c and d are any scalars, then

212121 , and ,, ,, wwvvuu wvu

2211 , uvuv uv

2211 , uvuv uv

21,cvcvc v

uvvu

wvuwvu

vvv dcdc

wvwv ccc

vv cddc


EXAMPLE 3 Operations on Vectors

Find each expression.a. v + w b. –2v c. 2v – w d. ||2v – w||

2 2 2,3 b. v

Solution 2,3 4,1 a. v w

Let v 2, 3 and w 4,1 .

2 4, 31 2, 4

22, 23 4, 6


EXAMPLE 3 Operations on Vectors

Solution continued 2 2 2,3 4,1 c. v w

2 8,5 d. v w

4,6 4,1

4 4 ,6 1

8,5

2v w 82 52

2v w 64 25 89


UNIT VECTORS

A vector of length 1 is a unit vector.

1v

v.

The unit vector in the same direction as v is given by


UNIT VECTORS IN i, j FORM

In a Cartesian coordinate plane, two important unit vectors lie along the positive coordinate axes:

1,0 and 0,1 ji

The unit vectors i and j are called standard unit vectors.


UNIT VECTORS IN i, j FORMEvery vector can be expressed in terms of i and j as follows:

21,vvv

ji

v

21

21

2121

1,00,1

,00,,

vv

vv

vvvv

A vector v from (0,0) to (v1,v2) can therefore be

represented in the form v = v1i + v2j with.2

221 vv v


VECTORS IN TERMS OFMAGNITUDE AND DIRECTION

Let is the smallest nonnegative angle that v makes with the positive x-axis. The angle is called the direction angle of v.

v v cosi sin j The formula

be a position vector and suppose

expresses a vector v in terms of its magnitude ||v|| and its direction angle .

21,vvv


EXAMPLE 6Writing a Vector with Given Length and Direction Angle

Write the vector of magnitude 3 that makes an angle of with the positive x-axis.

3

Solution

jiv

jiv

jiv

jivv

233

23

23

213

3sin

3cos3

sincos


APPLICATIONS OF VECTORS

If a system of forces acts on a particle, that particle will move as though it were acted on by a single force equal to the vector sum of the forces. This single force is called the resultant of the system of forces.

If the particle does not move, the resultant is zero and we say that the particle is in equilibrium.


EXAMPLE 8 Finding the Resultant

Solution

Find the magnitude and bearing of the resultant R of two forces F1 and F2, where F1 is a 50 lb force acting northward and F2 is a 40 lb force acting eastward.

First, set up a coordinate system.



Solution continued



Solution continued

The angle between R and the y-axis (north) is 90º −51.3º = 38.7º.

Therefore, the resultant R is a force of approximately 64.0 lbs in the directionN 38.7º E.

1 2011 Pearson Education, Inc. All rights reserved 1 2010 Pearson Education, Inc. All rights reserved 2011 Pearson Education, Inc. All rights reserved.

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