Transcript Pre-Calculus Chapter 6

Chapter 6
Additional Topics in Trigonometry
6.3 Vectors in the Plane
Objectives:
Represent vectors as directed line segments.
Write the component form of vectors.
Perform basic vector operations and represent
vectors graphically.
Write vectors as linear combinations of unit
vectors.
Find the direction angles of vectors.
Use vectors to model & solve real-life problems.
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Question….
How do we represent quantities that have both a
size (length or magnitude) and a direction?
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Answer: Vectors
 A vector is a quantity with both a size (magnitude)
and a direction.
(Note: A quantity with size only is called a scalar.)
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For Example….
 A ball flies through the air at a
certain speed and in a particular
direction.
 The speed and direction are the
velocity of the ball.
 The velocity is a vector quantity
since it has both a magnitude and a
direction.
 Examples of vector quantities:
displacement, velocity, force
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How Do We Represent Vectors?
A vector is represented by a directed line segment
PQ with initial point P and terminal point Q.
Q
P
 The arrow defines the direction.
Direction is found using Slope Formula.
 The length of the segment defines the magnitude.
Magnitude is found using Distance Formula.
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Vector Notation
 Vectors can be represented in different ways:
 Lower-case, boldface letters such as u, v,
and w.
 The letters of the endpoints of the directed
line segment with an arrow above.

v  PQ

 Magnitude is denoted by PQ .
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Equivalent Vectors
Equivalent Vectors:
Have the same direction (parallel).
Have the same magnitude.
v
u
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Example 1
 Let u be represented by the directed line segment from
P = (0, 0) to Q = (3, 2), and let v be represented by the
directed line segment from R = (1, 2) to S = (4, 4).
Show that u = v.
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Results for Example 1
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Vector in Standard Position
Standard Position:
 Initial point is the origin (0, 0).
 Represented uniquely by its
terminal point (u1, u2).
If initial point is not the origin:
y
 Can be rewritten as a vector in
standard position.
 This is called the component
form of the vector.
y
(u1, u2)
x
P (p1, p2)
Q (q1, q2)
x
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Component Form of a Vector
The component form of the vector with
Initial Point P = (x1, y1) and
Terminal Point Q = (x2, y2) is given by:

PQ  x2  x1 , y2  y1  v1 , v2  v
The magnitude (or length) of v is given by:
v 
x2  x1    y2  y1 
2
2
 v v
2
1
2
2
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Zero Vector and Unit Vector
 Zero Vector:
Both the initial point and the terminal point
lie at the origin.

Notation:
0  0, 0
 Unit Vector:
Magnitude of the vector is equal to 1.
Notation:
v 1
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Example 2
Find the component form and magnitude of the
vector v that has initial point (4, –7) and terminal
point (–1, 5).
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Equal Vectors
Two vectors u = <x1, y1> and v = <x2, y2>
are equal if and only if x1 = x2 and y1 = y2.
Example 1 (again):
Vector u from P = (0, 0) to Q = (3, 2), and
vector v from R = (1, 2) to S = (4, 4). Show
that u = v by writing each in component
form.
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Vector Operations
Let u = (x1, y1), v = (x2, y2), and let k be a scalar.
1. Scalar Multiplication
ku = (kx1, ky1)
2. Vector Addition
u + v = (x1+x2, y1+ y2)
3. Vector Subtraction
u  v = (x1  x2, y1  y2)
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Scalar Multiplication
The product of a vector and
a constant, k (“scalar”).
New vector is │k│ times as
long as v.
If k is positive, kv has
same direction as v.
If k is negative, kv has
opposite direction of v.
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Vector Addition
To add vectors u and v:
u
v
1. Place the initial point of v at the terminal point of u.
2. Draw the vector with the same initial point as u and
the same terminal point as v.
u
v
v
u
u+v
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Diagram for Vector Addition
 “Parallelogram Law” – The sum is the “resultant”
of the parallelogram.
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Vector Subtraction
To subtract vectors u and v:
u
v
1. Place the initial point of v at the initial point of u.
2. Draw the vector u – v from the terminal point of v
to the terminal point of u.
u
u
uv
v
v
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Definition of Vector Addition and
Scalar Multiplication
 Let u = <u1, u2> and v = <v1, v2> be vectors and
let k be a scalar.
The sum of u and v is the vector
The scalar multiple of k times u is the vector
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Example 3
 Let v = <–2, 5> and w = <3, 4>.
Find each of the following vectors.
a.
2v
b.
w–v
c.
v + 2w
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Graphical Representation
of Solutions
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Properties of Vector Addition and
Scalar Multiplication
 Let u, v, and w be vectors and let c and d be
scalars. Then the following properties are true.
1. u + v = v + u
6. (c + d)u = cu + du
2. (u + v) + w = u + (v + w) 7. c(u + v) = cu + cv
3. u + 0 = u
8. 1(u) = u
4. u + (–u) = 0
9. 0(u) = 0
5. c(du) = (cd)u
10. || cv || = | c | * || v ||
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The Unit Vector
A vector, u, with magnitude equal to 1 that has
the same direction as the given vector, v.
 To find u, divide v by its length.
 Note that u is a scalar multiple of v.
 The vector u is called a unit vector in the
direction of v.
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Example 4
Find a unit vector in the direction of v = <–2, 5>
and verify that the result has a magnitude of 1.
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Standard Unit Vectors
The unit vectors
i = <1, 0> and j = <0, 1>
are called the standard
unit vectors.
Note that the standard
unit vector i is not the
imaginary number
i  1
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Linear Combination
 Vector v = <v1, v2> can be written as
 The scalars v1 and v2 are called the horizontal
and vertical components of v.
 The vector sum v1i + v2j is called the linear
combination of the standard unit vectors i and j.
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Example 5
Let u be the vector with initial point (2, –5)
and terminal point (–1, 3). Write u as a
linear combination of the standard unit
vectors i and j.
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Example 6
 Let u = –3i + 8j and v = 2i – j.
Find 2u – 3v without converting the vectors
to component form.
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Direction Angles
The direction angle  of a vector v is the angle
formed by the positive half of the x-axis and the
ray along which v lies.
y
y
θ
v
θ
x
x
v
y
If v  x, y , then tan   .
x
y
(x, y)
v
x
y
x
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Example 7
 Find the direction angle of each vector:
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Example 8
 Find the component
form of the vector that
represents the velocity of
an airplane descending at
a speed of 100 miles per
hour at an angle of 30°
below the horizontal, as
shown.
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Homework 6.3
Worksheet 6.3
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