Ch 3
Vectors
Vectors
• What is the difference between a scalar
and a vector?
• A vector is a physical quantity that has
both magnitude and direction
• What are some examples of vectors
that we have used in this class?
Vector vs. Scalar
• State whether each of the following
quantities is a vector or a scalar:
Position
Velocity
Vector
Vector
Displacement
Vector
Distance
Acceleration
Vector
Speed
Scalar
Volume
Energy
Scalar
Scalar
Scalar
Force
Vector
Temperature
Scalar
Pressure
Scalar
Representing Vectors
• Remember that vectors have magnitude
AND direction.
90°
135°
45°
180°
A  6.8m @ 32.5
0°
Adding Vectors Graphically
• Pick a scale for your drawing.
• Draw the first vector starting at the origin.
• Place your protractor at the “Head” of the first
vector to make the correct angle.
• Draw the next vector such that it starts at the
head of the first vector.
• Continue to line each vector up head to tail.
• Draw the resultant vector.
• Measure its length for the magnitude and angle
for its direction.
Resultant Vector
• Resultant Vector is the sum of 2 or more
vectors.
• Drawn with a dashed line.
v2  1.5
v1  0.80
m
@ 90
s
m
@ 0
s
to tip of last vector
vR
Drawn from tail of first vector
• Vectors can be added graphically by
placing the “Tail” of one vector to the
“Head” of the other.
• The Resultant is the sum of components of two
or more vectors
– The resultant can be found by drawing a vector from
the origin to the head of the last vector
Adding Vectors Graphically
If you walked 6 blocks East and
then 4 blocks north
What is your displacement?

R  7.2 Blocks @33.6o
Adding Vectors Graphically
• When graphically adding vectors:
– The scale must not change
– The direction of the reference angle must not
change
Adding Vectors Graphically
• Does the order in which you add the
vectors matter?
• 1+2+3=6
• 3+2+1=6
• 2+1+3=6
Adding Vectors Graphically
• You walk 5m @ 0o and then turns to walk 6m @90o.
Finally, you turn to walk 8 m at 200°. What is your
displacement?
Addition is
commutative!
d R  4@ 120
Adding Vectors Graphically
• Multimedia
– Vector addition, order does not mater.
Adding Vectors Graphically
• Vectors are always
added head to tail
• Always measure the
angle from the +x axis.
• Vectors are express in
two parts
• Given vectors A and B,
find vector C.
A  2.5@ 45
B  4.50@150

C ?
C  4.5@120o
Adding Vectors Graphically
• Given vectors
A and B, find
vector C.
C=B +A
A  2.5@ 45
B  4.50@150

C ?
C  4.5@120o
Adding Vectors Graphically
• Vector 1:
300.0 m @ 0
• Vector 2
450.0 m @ 135 
• Vector 3
250.0 m @ 270
• What is the Resultant?
• D=A+B+C

D  70m @105o
Relative Velocity
Relative Velocity
Relative Velocity
Independence of Vectors
• Multimedia
• The river boat
• The plane and the wind
Independence of Vectors
A boat travels north at 8m/s
across an 80m wide river
which flows west at 5m/s .
The river is 80m wide
Independence of Vectors
• Perpendicular vector quantities are
independent of each other.
• For example in projectile motion
– Vx Velocity in the X-direction
– Vy Velocity in the Y-direction
Are independent of each other.
Trig Function Reminders
• Trig functions take angles
for input and give ratios
for their output.
Trig
Functions
• Inverse Trig functions
take ratios for input and
give angles for output.
Inverse Trig
Functions
Adding Force Vectors Analytically
cos  
opposite
sin  
Opposite
Hypontenuse

adjacent
Hypontenuse
adjacent
tan  
opposite
adjacent
o
tan ( )  
a
1
Components of Vectors
Finding the vector magnitude and direction
when you know the components.
A  Ax2  Ay2
tan  
Ay
Ax
   tan
1
Ay
Ax
Recall:  is measured
from the positive x axis.
Caution: Beware of the tangent function.
Always consider in which quadrant the vector lies when
dealing with the tangent function.
I
II
  tan 1 (5 /  8.66)
  30
  150
  tan 1 (5 / 8.66)
  30
5
5
-8.66
-8.66
8.66
-5
  tan 1 (5 /  8.66)
  30
  210
III
-5
8.66
  tan 1 (5 / 8.66)
  30
  330
IV
Adding Vectors Analytically
• Resolve each vector into its horizontal and
vertical components
• Add all of the vertical components together
• Add all of the horizontal components
together
• Draw a right triangle using the horizontal
and vertical resultants
Adding Vectors Analytically
Ax  A cos 
Ay  A sin 
A  4.5 N @30
By  B sin 
B  7 N @ 210
Bx  B cos 
C y  C sin 
C  6 N @150 Cx  C cos 

o

o

o
Magnitude
Angle
4.5N
7N
6N
---------------
30o
210o
150o
------------
R=7.55N
X component Y component
3.89N
-6.06N
-5.19N
Rx=-7.36N
2.25N
-3.5N
3.0N
Ry=1.7N
Angle =-13o+180o = 167o
Adding Vectors WS 17
• Analytically and Graphically add the
following vector sets.
•
•
•
•
v1
v2
v3
v4
17m/s @ 300
24m/s @ 170
24m/s @ 55o
19m/s @ 20o
Practice Problem WS6a #1

A  17 @ 300

m
s
C  24 @ 55
m
s
o
o
Magnitude
Angle
17
300
24
170
24
55
19
20
-------------- -----------R=22.7m/s

B  24 ms @170o

D  19 ms @ 20o
Rx
8.5
-23.6
13.76
17.85
16.51
Angle = 43.4o
Ry
-14.7
4.17
19.65
6.49
15.62
Multiplying a Vector by a Scalar
A
B
A
C
½A
A
B = 2A
C = -1/2 A
Adding “-” Vectors
 Add “negative” vectors
by keeping the same
magnitude but adding
180 degrees to the
direction of the original
vector.
C=A+B
D=A-B
D = A + (- B)
B
C
-B
A
D
Vector Concept questions
• What method is used to add vectors
graphically?
• How is the resultant vector affected if the
force vectors are added in a different
order?
• What is equilibrium?
Vector Concept questions
•
A vector is to be added graphically,
which, if any, of the following may you do
the first vector?
a)
b)
c)
d)
Rotate it
Move it
Lengthen it
Shorten it
Vector Concept questions
• What is the sum of three vectors that form
a triangle?
• If these vectors are forces, what does the
imply about the object the forces are
acting on?
Adding Vectors
• Graphically and Analytically add the
following vector sets.
– V1
5.2m/s @ 70
– V2
6.4m/s @ 210
– V1
– V2
10m/s @ 45
15m/s @ 135
Components of Vectors
• Vector resolution is the process of finding
the two component vectors.
Graphical Vector Quiz
On the first part of his flight, Jason flies his plane
5.0 miles due east ( = 5.0 miles @ 0).
He then turns and flies 10.0 miles North West (
10.0 miles @ 135).
Finally, he turns due south and flies 3.0 miles ( 3.0
miles @ 270).
What is his displacement from his takeoff point ?
Quiz Solution

C

B

R

A

R  4.5mi @117o
Blocks are 1 cm x 1 cm
Scale: 1 cm = 2 miles
• End Ch6 Vectors
Two Body
Probems
Adding Vectors Graphically
C  4.5@120o
Protractor
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Adding Vectors Graphically
• To find the magnitude of the resultant,
measure the length
• To find the direction of the resultant,
measure the angle.
– The direction is always measured counter
clockwise from the horizontal (east)
• Multimedia vector direction
Adding Vectors Analytically

A  7 N @ 45o

B  8 N @180o

C  6 N @ 270
o
Ax  A cos 
Ay  A sin 
Bx  B cos 
By  B sin 
Cx  C cos 
C y  C sin 

Ay
=45o
Ax
B=8
C=6
Add the yx components together
Compute the Resultant
Adding Vectors Analytically
Ax  A cos 
Ay  A sin 
A  7 N @ 45
By  B sin 
B  8 N @180
Bx  B cos 
C y  C sin 
C  6 N @ 270 Cx  C cos 



o
o
o
Magnitude
Angle
7N
45o
8N
180o
6N
270o
-------------------------R=3.22N
X
Y
4.95N
4.95N
-8N
0N
0N
-6N
Rx=-3.05N Ry=-1.05N
Angle =19o+180o = 199o
Adding Force Vectors Graphically
Add the following
3 vectors

A  7 N @ 45o

B  8 N @180o

C  6 N @ 270o

R  3.25 N @ 200
o