Calculus 10.2

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10.2 day 1 Vectors in the Plane
Mesa Verde National Park, Colorado
Greg Kelly, Hanford High School, Richland,
Washington
Photo by Vickie Kelly, 2003
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Warning Only some of this is review.
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Quantities that we measure that have magnitude
but not direction are called scalars.
Quantities such as force, displacement or
velocity that have direction as well as magnitude
are represented by directed line segments.
B
terminal point
The length is
A
initial point
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B
terminal point
A
initial point
A vector is represented by a directed line
segment.
Vectors are equal if they have the same length
and direction (same slope).
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y
A vector is in standard position if the initial
point is at the origin.
x
The component form of this vector is
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y
A vector is in standard position if the initial
point is at the origin.
x
The component form of this vector is
The magnitude (length) of
is
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The component form of
(-3,4)
P
is
(-5,2)
Q
v
(-2,-2)
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Then v is a unit vector.
If
is the zero vector and has no direction.
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Vector Operations
(Add the components.)
(Subtract the components.)
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Vector Operations
Scalar Multiplication
Negative (opposite)
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u
v
u v is the resultant vector.
uv
(Parallelogram law of addition)
v
u
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The angle between two vectors is given by
This comes from the law of cosines. See page 524
for the proof if you are interested.
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The dot product (also called inner product) is
defined as
Read u dot v
Example
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The dot product (also called inner product) is
defined as
This could be substituted in the formula for the
angle between vectors (or solved for theta) to
give
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Example
Find the angle between vectors u and v
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Application Example 7
A Boeing 727 airplane, flying due east at 500mph
in still air, encounters a 70-mph tail wind
acting in the direction of 60o north of east.
The airplane holds its compass heading due east
but, because of the wind, acquires a new ground
speed and direction. What are they?
N
E
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Application Example 7
A Boeing 727 airplane, flying due east at 500mph
in still air, encounters a 70-mph tail wind
acting in the direction of 60o north of east.
The airplane holds its compass heading due east
but, because of the wind, acquires a new ground
speed and direction. What are they?
N
E
u
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Application Example 7
A Boeing 727 airplane, flying due east at 500mph
in still air, encounters a 70-mph tail wind
acting in the direction of 60o north of east.
The airplane holds its compass heading due east
but, because of the wind, acquires a new ground
speed and direction. What are they?
N
v
60o
E
u
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Application Example 7
A Boeing 727 airplane, flying due east at 500mph
in still air, encounters a 70-mph tail wind
acting in the direction of 60o north of east.
The airplane holds its compass heading due east
but, because of the wind, acquires a new ground
speed and direction. What are they?
N
We need to find the magnitude and direction of
the resultant vector u v.
v
uv
E
u
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N
The component forms of u and v are
v
70
uv
E
u
500
Therefore
and
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N
538.4
6.5o
E
The new ground speed of the airplane is about
538.4 mph, and its new direction is about 6.5o
north of east.
p