Transcript Notes 7.6 - TeacherWeb

Chapter 7
Applications of
Trigonometric
Functions
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All rights reserved
© 2010
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SECTION 7.6
The Dot Product
OBJECTIVES
1
2
3
4
5
6
Define the dot product of two vectors.
Find the angle between two vectors.
Define orthogonal vectors.
Find the projection of a vector onto another
vector.
Decompose a vector into two orthogonal
vectors.
Use the definition of work.
THE DOT PRODUCT
For two vectors v  v1 , v2 and w  w1 , w2 ,
the dot product of v and w, denoted v • w, is
defined as
v  w  v1 , v2  w1 , w2  v1w1  v2 w2 .
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EXAMPLE 1
Finding the Dot Product
Find the dot product v • w.
a. v  2, 3 and w  3, 4
b. v  3i  5 j and w  2i  3j
Solution
a. v  w  2, 3  3, 4
 2 3  34   6
b. v  w  3i  5 j 2i  3j
 3, 5  2, 3
 32   5 3  9
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THE DOT PRODUCT AND
THE ANGLE BETWEEN TWO VECTORS
If  (0° ≤  ≤ 180°) is the angle between two
nonzero vectors v and w, then
vw
v  w  v w cos  or cos  
v w
 vw 
.
and   cos 

v
w


1
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EXAMPLE 2
Finding the Dot Product
If v and w are two vectors of magnitudes 5 and
7, respectively, and the angle between them is
75º, find v · w. Round the answer to the nearest
tenth.
Solution
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PARALLEL VECTORS
Two vectors v and w are parallel if there is a
nonzero scalar, c, so that v = cv.
The angle θ between parallel vectors is either
0º or 180º.
So v and w are parallel if v · w = ±||v||||w||.
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EXAMPLE 4
Finding the Angle Between Two Vectors
Find the angle θ (in degrees) between the vectors
v = 2i + 3j and w = −3i + 4j. Round the answer
to the nearest tenth of a degree.
Solution
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EXAMPLE 4
Finding the Angle Between Two Vectors
Solution continued
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PROPERTIES OF THE DOT PRODUCT
If u, v, and w are vectors and c is a scalar,
then
1. u  v  v  u.
2. u  v  w   v  u  u  w.
3. 0  v  0.
4. v  v  v .
2
5. cu   v  cu  v   u  cv .
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ORTHOGONAL VECTORS
Two vectors v and w are orthogonal
(perpendicular) if and only if v · w = 0.
Because 0 · w = 0 for any vector w, it
follows from the definition that the zero
vector is orthogonal to every vector.
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EXAMPLE 7
Orthogonal Vectors
Find the scalar c so that the vectors v = 3i + 2j
and w = 4i + cj are orthogonal.
Solution
The vectors v = 3i + 2j = 3, 2 and
w = 4i + cj = 4, c are orthogonal if and only if
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PROJECTION OF A VECTOR
Let OP and OQ be representations of the
nonzero vectors v and w, respectively. The
projection of OP in the direction of OQ is the
directed line segment OR , where R is the foot of
the perpendicular from P to the line containing
OQ.
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PROJECTION OF A VECTOR
The vector represented by OR is called the
vector projection of v onto w and is denoted
by projwv.
Let θ (0º ≤ θ ≤ 180º) be the measure of the
angle between nonzero vectors v and w. The
number ||v|| cos θ is called the
scalar projection of v onto w.
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VECTOR AND SCALAR PROJECTIONS
OF v ONTO w
Let v and w be two nonzero vectors and let θ
(0º ≤ θ ≤ 180º) be the angle between them.
The vector projection of v onto w is
 vw 
w.
projw v  
 w2


The scalar projection of v onto w is
vw
v cos  
.
w
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DECOMPOSITION OF A VECTOR
Let v and w be two nonzero vectors. The vector v
can be written in the form
v = v1 + v2,
where v1 is parallel to w and v2 is orthogonal to w.
We have
 vw 
w and v  v  v .
v1  projw v  
2
1
 w2


The expression v = v1 + v2 is called the
decomposition of v with respect to w. The vectors
v1 and v2 are called components of v.
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DEFINITION OF WORK
The work W done by a constant force F in
moving an object from a point P to a point Q
is defined by
where θ is the angle between F and PQ.
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EXAMPLE 10 Computing Work
A child pulls a wagon along level ground, with a
force of 40 pounds along the handle, which
makes an angle of 42º with the horizontal. How
much work has she done by pulling the wagon
150 feet?
Solution
W  F  PQ
W  F PQ cos
W   40 150  cos 42º
W  4458.87 foot-pounds
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