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8
Applications
of
Trigonometry
Copyright © 2009 Pearson Addison-Wesley
8.3-1
8 Applications of Trigonometry
8.1 The Law of Sines
8.2 The Law of Cosines
8.3 Vectors, Operations, and the Dot Product
8.4 Applications of Vectors
8.5 Trigonometric (Polar) Form of Complex
Numbers; Products and Quotients
8.4 De Moivre’s Theorem; Powers and Roots of
Complex Numbers
8.5 Polar Equations and Graphs
8.6 Parametric Equations, Graphs, and
Applications
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8.3 Vectors, Operations, and the
Dot Product
Basic Terminology ▪ Algebraic Interpretation of Vectors ▪
Operations with Vectors ▪ Dot Product and the Angle Between
Vectors
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Basic Terminology
 Scalar: The magnitude of a quantity. It can be
represented by a real number.
 A vector in the plane is a directed line segment.
 Consider vector OP
O is called the initial point
P is called the terminal point
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Basic Terminology
 Magnitude: length of a vector, expressed as
|OP|
 Two vectors are equal if and only if they have
the same magnitude and same direction.
Vectors OP and PO
have the same
magnitude, but opposite
directions.
|OP| = |PO|
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Basic Terminology
A=B
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C=D
A≠E
A≠F
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Sum of Two Vectors
 The sum of two vectors is also a vector.
 The vector sum A + B is called the resultant.
Two ways to represent the sum
of two vectors
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Sum of Two Vectors
 The sum of a vector v and its opposite –v has
magnitude 0 and is called the zero vector.
 To subtract vector B
from vector A, find
the vector sum A +
(–B).
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Scalar Product of a Vector
 The scalar product of a real number k and a
vector u is the vector k ∙ u, with magnitude |k|
times the magnitude of u.
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Algebraic Interpretation of
Vectors
 A vector with its initial point at the origin is called
a position vector.
 A position vector u with its endpoint at the point
(a, b) is written
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Algebraic Interpretation of
Vectors
 The numbers a and b are the horizontal
component and vertical component of
vector u.
 The positive angle
between the x-axis and a
position vector is the
direction angle for the
vector.
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Magnitude and Direction Angle
of a Vector a, b
The magnitude (length) of a vector u = a, b
is given by
The direction angle θ satisfies
where a ≠ 0.
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Example 1
FINDING MAGNITUDE AND DIRECTION
ANGLE
Find the magnitude and direction angle for u = 3, –2.
Magnitude:
Direction angle:
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Example 1
FINDING MAGNITUDE AND DIRECTION
ANGLE (continued)
Graphing calculator solution:
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Horizontal and Vertical Components
The horizontal and vertical components,
respectively, of a vector u having magnitude
|u| and direction angle θ are given by
or
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FINDING HORIZONTAL AND VERTICAL
COMPONENTS
Example 2
Vector w has magnitude 25.0 and direction angle
41.7°. Find the horizontal and vertical components.
Horizontal component: 18.7
Vertical component: 16.6
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Example 2
FINDING HORIZONTAL AND VERTICAL
COMPONENTS
Graphing calculator solution:
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Example 3
WRITING VECTORS IN THE FORM a, b
Write each vector in the figure in
the form a, b.
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Properties of Parallelograms
1. A parallelogram is a quadrilateral whose
opposite sides are parallel.
2. The opposite sides and opposite angles
of a parallelogram are equal, and
adjacent angles of a parallelogram are
supplementary.
3. The diagonals of a parallelogram bisect
each other, but do not necessarily bisect
the angles of the parallelogram.
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Example 4
FINDING THE MAGNITUDE OF A
RESULTANT
Two forces of 15 and 22 newtons act on a point in the
plane. (A newton is a unit of force that equals .225 lb.)
If the angle between the forces is 100°, find the
magnitude of the resultant vector.
The angles of the parallelogram
adjacent to P measure 80°
because the adjacent angles of a
parallelogram are supplementary.
Use the law of cosines with ΔPSR
or ΔPQR.
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Example 4
FINDING THE MAGNITUDE OF A
RESULTANT (continued)
The magnitude of the resultant vector is about
24 newtons.
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Vector Operations
For any real numbers a, b, c, d, and k,
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Example 5
PERFORMING VECTOR OPERATIONS
Let u = –2, 1 and v = 4, 3. Find the following.
(a) u + v = –2, 1 + 4, 3 = –2 + 4, 1 + 3 = 2, 4
(b) –2u = –2 ∙ –2, 1 = –2(–2), –2(1) = 4, –2
(c) 4u – 3v = 4 ∙ –2, 1 – 3 ∙ 4, 3
= –8, 4 –12, 9
= –8 – 12, 4 – 9 = –20,–5
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Unit Vectors
 A unit vector is a vector that has magnitude 1.
i = 1, 0
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j = 0, 1
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Unit Vectors
Any vector a, b can be expressed in the form
ai + bj using the unit vectors i and j.
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Dot Product
The dot product (or inner product) of the two
vectors u = a, b and v = c, d is denoted
u ∙ v, read “u dot v,” and is given by
u ∙ v = ac + bd.
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Example 6
FINDING DOT PRODUCTS
Find each dot product.
(a) 2, 3 ∙ 4, –1 = 2(4) + 3(–1) = 5
(b) 6, 4 ∙ –2, 3 = 6(–2) + 4(3) = 0
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Properties of the Dot Product
For all vectors u, v, and w and real number
k,
(a) u ∙ v = v ∙ u
(b) u ∙ (v + w) = u ∙ v + u ∙ w
(c) (u + v) ∙ w = u ∙ w + v ∙ w
(d) (ku) ∙ v = k(u ∙ v) = u ∙ kv
(e) 0 ∙ u = 0
(f) u ∙ u = |u|2
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Geometric Interpretation of
the Dot Product
If θ is the angle between the two nonzero
vectors u and v, where 0° ≤ θ ≤ 180°, then
or
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Example 7
FINDING THE ANGLE BETWEEN TWO
VECTORS
Find the angle θ between the two vectors u = 3, 4
and v = 2, 1.
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Dot Products
For angles θ between 0° and 180°, cos θ is positive, 0,
or negative when θ is less than, equal to, or greater
than 90°, respectively.
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Note
If a ∙ b = 0 for two nonzero vectors a
and b, then cos θ = 0 and θ = 90°.
Thus, a and b are perpendicular or
orthogonal vectors.
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