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Chapter 9
Vectors and Oblique Triangles
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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.
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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.

A
Write 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 _______________________.
C
We write:
A
B
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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.
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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
B
A
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 2A + 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 2A - B
B
A
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Label vectors appropriately!
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
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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.
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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
Vy
63.5°
x
0
Vx
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To find the x- and y-components of V, we will use right
triangle trigonometry.
y
V=13.8
x-component
y-component
Vx
cos  63.5  
V
Vy
sin  63.5  
V
V
Vy
63.5°
x
0
Vx
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To Resolve a Vector Into its x- and ycomponents:
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 Vx = V cos 
3.
Calculate the y-component by Vy = V sin 
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Example:
Find the x- and y-components of the given vector by use of the trig
functions.
1) 9750 N,  = 243.0°
y
0
x
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Example:
Find the x- and y-components of the given vector by
use of the trig functions.
2) 16.4 cm/s2,  = 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.
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From the text:
P. 262 # 28
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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 A are Ax = 735 and Ay = 593, find
the magnitude of A and the angle it makes with the x axis.
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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 Rx
Add
the y-components together to get Ry
Use
the Pythagorean theorem to find the magnitude of the resultant.
R
 Rx    Ry 
2
2
Use
the inverse tangent function to help find the angle that gives the
direction of the resultant.
ref  tan
1
Ry
Rx
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To determine the measure of angle , you need to know
the quadrant in which R lies.
If R lies in
Quadrant I  Rx  0& Ry  0  :    ref
Quadrant II  Rx  0& Ry  0  :   180   ref
Quadrant III  Rx  0& Ry  0  :   180   ref
Quadrant IV  Rx  0& Ry  0  :   360   ref
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Example
Find the resultant of three vectors A, B, and C, such that
A  6.34,  A  29.5; B  4.82,  B  47.2; and C  5.52, C  73.0
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From the text: p. 267 # 8, 28
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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
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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.
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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)
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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)
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Case 3:
Two sides and the included angle (SAS)
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Case 4:
Three sides (SSS)
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C
b
a
B
A
c
The Law of Sines
a
b
c


sin A sin B sin C
Three equations for the price of one!
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Solving Case 1: ASA or SAA
Solve the triangle:
A  35.0, B  15.0, c  5.00
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Solving Case 1: ASA or SAA
A  40.0, B  60.0, a  4.00
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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
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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.
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SSA --- The Ambiguous Case
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If a  h  b sin A, then side a is not
sufficiently long enough to form a triangle.
No Triangle
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If a  h  b sin A, then
side a is just long enough to form a right triangle.
One Right Triangle
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If h  b sin A  a and a  b, two
distinct triangles can be formed from the given
information.
Two Triangles
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One Triangle
If a  b, only one triangle can be
formed.
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a  3.0, b  2.0, A  40
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Continued from above
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a  6.0, b  8.0, A  35
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Continued from above
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a  1.0, b  2.0, A  50
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Making fairly accurate sketches can help you to determine the
number of solutions.
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Example: Solve ABC where A = 27.6, a =112, and c = 165.
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Continued from above
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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.
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C
b
a
B
A
The Law of Cosines
c
a  b  c  2bc cos A
2
2
2
b  a  c  2ac cos B
2
2
2
c  a  b  2ab cos C
2
2
2
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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  a  b  2ab cos C
2
2
2
2ab cos C  a  b  c
2
2
a b c
cos C 
2ab
2
2
2
2
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.
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Continued from above
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Solving Case 3: SAS
Example: Solve ABC where b = 16.4, c = 10.6, and A = 128.5.
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Continued from above
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Guidelines for Solving Case 4: SSS
When given three sides, follow these steps:
1.
Use the Law of Cosines to find the LARGEST
ANGLE *(opposite the largest side).
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
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When to use what……
(Let bold red represent the given info)
SAS
AAS
ASA
Be careful!!
May have 0, 1, or
2 solutions.
SSS
SSA
Use Law of Sines
Use Law of Cosines
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To nearest tenth of a mile.
To nearest tenth of a degree.
To nearest minute.
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Continued from above
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