Graphical
Analytical
Component Method
VECTOR ADDITION
MATH STUFF

What does an ordered pair mean in math?
Ex:(2,3)
VECTORS

Quantities having both magnitude and direction




Magnitude: How much (think of it as the length of the line)
Direction: Which way is it pointing?
Can be represented by an arrow-tipped line segment
Examples:




Velocity
Acceleration
Displacement
Force
QUESTION:

Compare the two vectors. What makes them
different
ANSWER
Direction
 The magnitude or length is exactly the same

VECTOR TERMINOLOGY


Two or more vectors acting on the same point are said
to be concurrent vectors.
The sum of 2 or more vectors is called the resultant
(R).



A single vector that can replace concurrent vectors
Any vector can be described as having both x and y
components in a coordinate system.
The process of breaking a single vector into its x and y
components is called vector resolution.
MORE VECTOR TERMINOLOGY
Vectors are said to be in equilibrium if their
sum is equal to zero.
 A single vector that can be added to others to
produce equilibrium is call the equilibrant (E).

 Equal
to the resultant in magnitude but opposite in
direction.
E+R=0
E=-R
E=5N
at 180 °
R=5N
at 0°
WHAT IS THE RESULTANT OF THE FOLLOWING
VECTORS?
E= 10 N at 0 degrees
 R = 20 N at 0 degrees

ANSWER

30 N at 0 degrees
QUESTION
20 N at 45 degrees
 10 N at 225 degrees


Do not get freaked out by the angles, Think
about it for a second.
ANSWER

10 N at 45 degrees
USING THE GRAPHICAL METHOD OF VECTOR
ADDITION:


Vectors are drawn to scale and the resultant is
determined using a ruler and protractor.
Vectors are added by drawing the tail of the second
vector at the head of the first (tip to tail method).


The order of addition does not matter.
The resultant is always drawn from the tail of the first
to the head of the last vector.
EXAMPLE PROBLEM
A 50 N force at 0° acts concurrently with a 20 N force at 90°.
R


R and  are equal on each diagram.
ADDING VECTORS!
QUESTION: ADD THESE TWO VECTORS
TOGETHER
a
b
ANSWER
b
a
R= a+b
MOTION APPLICATIONS
Perpendicular vectors act independently of one
another.
 In problems requesting information about
motion in a certain direction, choose the vector
with the same direction.

EXAMPLE PROBLEM:
MOTION IN 2 DIMENSIONS
 A boat heads east at 8.00 m/s across a river
flowing north at 5.00 m/s.
 What
is the resultant velocity of the boat?
A BOAT HEADS EAST AT 8.00 M/S ACROSS
A RIVER FLOWING NORTH AT 5.00 M/S.
5.00 m/s N
8.00 m/s E
River width
WHAT IS THE RESULTANT VELOCITY OF
THE BOAT?
Draw to scale and
measure.
5.00 m/s N
8.00 m/s E
R = 9.43 m/s at 32°
ADVANTAGES AND DISADVANTAGES
OF THE GRAPHICAL METHOD



Can add any number of
vectors at once
Uses simple tools
No mathematical
equations needed



Must be correctly draw to
scale and at appropriate
angles
Subject to human error
Time consuming
SOLVING VECTORS USING THE
ANALYTICAL METHOD
 A rough sketch of the vectors is drawn.
 The resultant is determined using:
 Algebra
 Trigonometry
 Geometry
QUICK REVIEW
Right Triangle
c is the hypotenuse
c2 = a2 + b2
B
c
sin = o/h
a
C
A
b
cos = a/h
A + B + C = 180°
B = 180° – (A + 90°)
tan A = a/b
tan B = b/a
tan = o/a
THESE LAWS WORK FOR
ANY TRIANGLE.
A + B + C = 180°
C
Law of sines:
a
sin A
b
a
B
=
b
= c
sin B
sin C
A
c
Law of cosines:
c2 = a2 + b2 –2abCos C
USE THE LAW OF:

Sines when you know:


2 angles and an opposite
side
2 sides and an opposite
angle

Cosines when you know:

2 sides and the angle
between them
FOR RIGHT TRIANGLES:
Draw a tip to tail sketch first.
 To determine the magnitude of the resultant

 Use

the Pythagorean theorem.
To determine the direction
 Use
the tangent function.
TO ADD MORE THAN TWO VECTORS:
Find the resultant for the first two vectors.
 Add the resultant to vector 3 and find the new
resultant.
 Repeat as necessary.

ADVANTAGES AND DISADVANTAGES
OF THE ANALYTICAL METHOD



Does not require
drawing to scale.
More precise answers
are calculated.
Works for any type of
triangle if appropriate
laws are used.



Can only add 2 vectors at
a time.
Must know many
mathematical formulas.
Can be quite time
consuming.
SOLVING VECTOR PROBLEMS
USING THE COMPONENT METHOD



Each vector is replaced by 2 perpendicular vectors
called components.
Add the x-components and the y-components to find
the x- and y-components of the resultant.
Use the Pythagorean theorem and the tangent
function to find the magnitude and direction of the
resultant.
VECTOR RESOLUTION
y = h sin 
h
y
x = h cos 

x
-+
++
--
+-
COMPONENTS OF FORCE:
y
x
QUESTION

What are the components of the following force
25N @ 12 degrees North of West
ANSWER
West is 180 degrees to 12 degrees north of
west is 168 degrees
 The X component is -24.45N
 The Y component is 5.20N
 You can confirm you answer –X and +Y would
be found in the second quadrant on a graph so
this answer makes sense

EXAMPLE:
x
6 N at 135°
5 N at 30°
y
5 cos 30° = +4.33
5 sin 30° = +2.5
6 cos 45 ° = - 4.24
6 sin 45 ° = + 4.24
+ 0.09
R = (0.09)2 + (6.74)2
= 6.74 N
 = arctan 6.74/0.09
= 89.2°
+ 6.74
TANGENT FUNCTION

The tangent function has 2 places that it is not
defined (you get an error on your calculator)
 90

degrees and 270 degrees
The x and y components tell you the angle
range
Angle Range
Operation
0 to 90
Nothing special needed
90 to 180
Add 180 degrees
180 to 270
Add 180 degrees
270 to 360
Add 360 degrees
QUESTION: CRITICAL THINKING

My X component was negative and my y
component was negative as well. My calculator
told me that my answer was 22 degrees. What
is my true angle?
ANSWER

My evidence:
 Negative
X
 Negative Y
We are in quadrant three (between 180
degrees and 270)
 I got 22 degrees, so I must take 180+22 to get
202 degree as my angle! Using my tangent
rules

Solve the following problem
using the component method.
6 km at 120
10 km at 30
ADDING VECTORS
Vector
X component
Y component
6 km @ 120 degrees
6 km * cos(120) =
6 km * sin(120) =
10 km @ 30 degrees
10 km * cos (30) =
10 km * sin(30) =
Add them together
To find the magnitude: pythagorean theorum
To find the direction:
1. Take into account if either X or y is + or –
2. Use any trig function SOH CAH TOA to find angle
CRITICAL THINKING QUESTION 2
I get a positive x and a negative y component
when I add them together.
 What degree range is my angle in?

ANSWER
X is positive so that can only mean either
quadrant 1 or 4
 Y is negative so that means you have to get
quadrant 4 as your answer
 270 to 360 degrees

NOTES
Make sure that all angles are measured from
the x axis (0 degrees)
 Report both the magnitude and the direction
otherwise the vector is wrong!
 Keep track of signs, They give you a clue to
where the angle of the vector actually is.

ADDING TWO VECTORS
Vector
X component
12 N @ 135 degrees
15 N @ 200 degrees
Resultant X and Y
Find the resultant magnitude:
Find the resultant direction:
Y component