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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
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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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4
5
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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
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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.
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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
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EQUIVALENT VECTORS
Two vectors having the same length and same direction are called equivalent vectors.
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EQUIVALENT VECTORS
Equivalent vectors are regarded as equal even though they may be located in different positions.
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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.
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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
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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
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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
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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.
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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
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EXAMPLE 1 Geometric Vectors
Solution continuedb. 2v – u + w
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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.
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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.
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EXAMPLE 8 Finding the Resultant
Solution continued
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EXAMPLE 8 Finding the Resultant
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.