Transcript Document
Chapter 9
Vectors and Oblique Triangles
1
9.1 An Introduction to Vectors
A
vector
quantity is one that
has ____________ as well as _________________
.
For example, velocity describes the direction of the motion as well as the magnitude (the speed).
A scalar quantity is one that has ___________ but no ______________
.
Some examples of scalar quantities are speed, time, area, mass.
2
Representing Vectors
In most textbooks, vectors are written in
boldface capital
letters. The scalar magnitude is written in lightface
italic
type.
So,
B
is understood to represent a vector quantity, having magnitude and direction, while
B
is understood to be a scalar quantity, having magnitude but no direction. When handwriting a vector, place an arrow over the letter to represent a vector. Write
A
A
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Geometrically, vectors are like
directed line segments
. Each vector has an
initial point
and and a
terminal point
.
• Q Terminal Point Initial Point P • Sometimes, vectors are expressed using the initial and terminal points.
PQ
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Two vectors are equal if they have the same _____________ and the same _______________________.
We write:
A B C
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Addition of Vectors (Two Methods)
The sum of any number of vectors is called the ____________________________, usually represented as ______.
Two common ways of adding vectors graphically are the
POLYGON METHOD,
and
PARALLELOGRAM METHOD
. 6
Polygon Method
To add vectors using the polygon method, position vectors so that they are
tail (dot) to head (arrow)
. The
resultant is the vector from the initial point (tail) of the first vector to the terminal point (head) of the second
. When you move the vector(s), make sure that the
magnitude and direction remain unchanged
! We use graph paper or a ruler and protractor to do this. Example : Add
A
+
B B A
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Polygon Method (cont.)
Vector addition is ________________________, which means that the order in which you add the vectors will not affect the sum.
Example : Add
B
+
A B A
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Polygon Method (cont.)
This method can be used to add three or more vectors.
Example : Add
A
+
B
+
C A B C
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Parallelogram Method
To add
two
vectors using the parallelogram method, position vectors so that they are
tail to tail (dot to dot)
, by letting the two vectors form the sides of a parallelogram. The
resultant is the diagonal of the parallelogram
. The initial point of the resultant is the same as the initial points of each of the vectors being added.
B A
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Scalar Multiplication
If
n
is a scalar number (no direction) and
A
is a vector, then nA is a vector that is in the same direction as
A
but whose magnitude is
n
times greater than
A
.
(Graphically, we draw this vector n times longer than A.)
Example : Add 2
A
+
B B A
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Subtraction of Vectors
Subtraction of vectors is accomplished by
adding the opposite
.
A
B = A + (
B)
where –
B
is the vector with same magnitude as
B
but opposite direction.
Example : Find 2
A
-
B B A
Label vectors appropriately!
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Force, velocity, and displacement are three very important vector quantities.
Force
is expressed with magnitude (in Newtons) and direction (the angle at which it acts upon an object).
Velocity
is expressed with magnitude (speed) and direction (angle or compass direction).
Displacement
is expressed with magnitude (distance) and direction (angle or compass direction).
Do classwork: Representing Vectors Graphically 13
9.2 – 9.3 Components of Vectors
Any vector can be replaced by two vectors which, acting together, duplicate the effect of the original vector. They are called
components
of the vector. The components are usually chosen perpendicular to each other. These are called
rectangular components
. The process of finding these components of a vector is called
resolving the vector into its components
. 14
We will resolve a vector into its x- and y-components by placing the initial point of the vector at the origin of the rectangular coordinate plane and giving its direction by an angle in standard position.
y
V=13.8
Vector
V
, of magnitude 13.8 and direction 63.5
°, and its components directed along the axes.
V V y
0 63.5
°
V x
x
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To find the x- and y-components of
V
, we will use right triangle trigonometry.
y
V y V
0 63.5
°
V x
V=13.8
x
x-component
V x V
y-component
V y V
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To Resolve a Vector Into its x- and y components:
1.
Place vector
V
with initial point at origin such that its direction is given by an angle in standard position.
2.
Calculate the x-component by
V x
= V cos 3.
Calculate the y-component by
V y
= V sin 17
Example:
Find the x- and y-components of the given vector by use of the trig functions.
1) 9750 N, = 243.0
°
y x
0 18
Example:
Find the x- and y-components of the given vector by use of the trig functions.
2) 16.4 cm/s 2 , = 156.5
°
y x
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A cable exerts a force of 558 N at an angle of 47.2
° with the horizontal. Resolve this into its horizontal and vertical components.
20
From the text: P. 262 # 28 21
Vector Addition by Components
We can use this idea of vector components to find the resultant of any two perpendicular vectors.
Example : If the components of vector the magnitude of
A A
are
A x
= 735 and
A y
= 593, find and the angle it makes with the
x
axis. 22
Example : Add perpendicular vectors
A
and
B
, given A = 4.85 and B =6.27
Find the magnitude and the angle that the resultant makes with vector
A
.
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Adding Non-Perpendicular Vectors
Place each vector with its tail at the origin Resolve each vector into its x- and y-components Add the x-components together to get R x Add the y-components together to get R y Use the Pythagorean theorem to find the magnitude of the resultant.
R
x
2
y
2 Use the inverse tangent function to help find the angle that gives the direction of the resultant.
ref
tan 1
R y R x
25
To determine the measure of angle , you need to know the
quadrant
in which R lies.
If R lies in Quadrant I Quadrant II Quadrant III Quadrant IV
R x R x
R x R x
0 & 0 & 0 & 0 &
R y R y R y R y
0 : 0 :
ref
180 180 360
ref
ref ref
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Example
Find the resultant of three vectors A, B, and C, such that
A
6.34,
A
B
4.82,
B
and C
5.52,
C
73.0
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From the text: p. 267 # 8, 28 29
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9.5 - 9.6 The Law of Sines and The Law of Cosines
In this section, we will work with
oblique triangles
triangles that do
NOT
contain a right angle
.
An oblique triangle has either: three acute angles or two acute angles and one obtuse angle 31
Every triangle has 3 sides and 3 angles. To
solve a triangle
means to find the lengths of its sides and the measures of its angles. To do this, we need to know at least three of these parts, and at least one of them must be a side.
32
Here are the
four possible combinations
of parts: 1. Two angles and one side (ASA or SAA) 2. Two sides and the angle opposite one of them (SSA) 3. Two sides and the included angle (SAS) 4. Three sides (SSS) 33
Case 1: Two angles and one side (ASA or SAA)
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Case 2: Two sides and the angle opposite one of them (SSA)
35
Case 3: Two sides and the included angle (SAS)
36
Case 4: Three sides (SSS)
37
A C b a B c
The Law of Sines
a
sin
A
b
sin
B
c
sin
C Three equations for the price of one!
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Solving Case 1: ASA or SAA
Solve the triangle:
A
B
c
5.00
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Solving Case 1: ASA or SAA
A
B
a
4.00
40
Example using Law of Sines
A ship takes a sighting on two buoys. At a certain instant, the bearing of buoy A is N 44.23
° W , and that of buoy B is N 62.17
° E . The distance between the buoys is 3.60 km , and the bearing of B from A is N 87.87
° E . Find the distance of the ship from each buoy.
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Continued from above 42
Solving Case 2: SSA
In this case, we are given two sides and an angle opposite.
This is called the
AMBIGUOUS CASE
. That is because it may yield
no solution
,
one solution
, or
two solutions
, depending on the given information. 43
SSA --- The Ambiguous Case
44
No Triangle
sufficiently long enough to form a triangle.
45
One Right Triangle
46
Two Triangles
distinct triangles can be formed from the given information. 47
One Triangle
formed. 48
a
3.0,
b
2.0,
A
40
49
Continued from above 50
a
6.0,
b
8.0,
A
51
Continued from above 52
a
1.0,
b
2.0,
A
53
Making fairly accurate sketches can help you to determine the number of solutions.
54
Example : Solve ABC where A = 27.6
, a =112, and c = 165. 55
Continued from above 56
To deal with Case 3 (SAS) and Case 4 (SSS), we do not have enough information to use the Law of Sines.
So, it is time to call in the
Law of Cosines
.
57
A b c C a B
The Law of Cosines
a
2
b
2
c
2
2
bc
cos
A b
2
a
2
c
2
2
ac
cos
B c
2
a
2
b
2
2
ab
cos
C
58
Using Law of Cosines to Find the Measure of an Angle
*To find the angle using Law of Cosines, you will need to solve the Law of Cosines formula for CosA, CosB, or CosC.
For example, if you want to find the measure of angle C, you would solve the following equation for CosC:
c
2
a
2
b
2
2
ab
cos
C
2
ab
cos
C
a
2
b
2
c
2 cos
C
a
2
b
2
c
2 2
ab To solve for C, you would take the cos -1 of both sides.
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Guidelines for Solving Case 3: SAS
When given two sides and the included angle, follow these steps: 1.
Use the Law of Cosines to find the third side.
2.
Use the Law of Cosines to find one of the remaining angles.
{
You could use the Law of Sines here, but you must be careful due to the ambiguous situation. To keep out of trouble, find the
SMALLER
of the two remaining angles (It is the one opposite the shorter side.)}
3.
Find the third angle by subtracting the two known angles from 180 .
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Solving Case 3: SAS
Example : Solve ABC where a = 184, b = 125, and C = 27.2
.
61
Continued from above 62
Solving Case 3: SAS
Example : Solve ABC where b = 16.4, c = 10.6, and A = 128.5
. 63
Continued from above 64
Guidelines for Solving Case 4: SSS
When given three sides, follow these steps: 1.
Use the
Law of Cosines ANGLE
to find the *(opposite the largest side).
LARGEST
2.
Use the
Law of Sines
to find either of the two remaining angles.
3.
Find the third angle by subtracting the two known angles from 180 .
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Solving Case 4: SSS
Example : Solve ABC where a = 128, b = 146, and c = 222.
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Continued from above 68
When to use what……
(Let bold red represent the given info) AAS Be careful!! May have 0, 1, or 2 solutions .
SSA
Use Law of Sines
ASA SAS SSS
Use Law of Cosines
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70
To nearest minute.
To nearest tenth of a mile.
To nearest tenth of a degree.
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Continued from above 72