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 AccelerationVelocity
SpeedDisplacement
Distance
Force
Energy
Temperature
Volume
Vector Vector Scalar
Scalar Scalar
Scalar Scalar
Vector
Vector Vector
Pressure
Scalar
Representing Vectors
• Remember that vectors have magnitude AND direction.
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@32.5
0°
90°
180°
135° 45°
1 2 3
4 5 6
7 8
9 1
0 11 12 13 14 15 16 17 18 19 20
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.2 1.5 @0
mv
s
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1 0.80 @90m
vs
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Rv��������������
Drawn from tail of first vector
to tip of last vector
• 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 GraphicallyIf you walked 6 blocks East and then 4 blocks north
What is your displacement?
7.2 @33.6oR Blocks
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!
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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.
4.50@150B
?C
2.5@ 45A
4.5@120oC
Adding Vectors Graphically• Given vectors
A and B, find vector C.
2.5@ 45A C = B + A
4.50@150B
?C
4.5@120oC
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
70 @105oD m
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.
• Inverse Trig functions take ratios for input and give angles for output.
Trig Functions
Inverse Trig Functions
Adding Force Vectors Analytically
sinOpposite
Hypontenuse
cosadjacent
Hypontenuse
tanopposite
adjacent
hypo
tenus
e
adjacent
opposite
1tan ( )o
a
Components of Vectors
Finding the vector magnitude and direction when you know the components.
Recall: is measured from the positive x axis.
22yx AAA
1tan tany y
x x
A A
A A
Caution: Beware of the tangent function.
Always consider in which quadrant the vector lies when dealing with the tangent function.
III
III IV
8.66
5
1tan (5 / 8.66) 30
-8.66
5
1tan (5 / 8.66)
30
-8.66
-5
1tan ( 5 / 8.66)
30
150
210
8.66
-5
1tan ( 5 / 8.66) 30
330
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
Magnitude Angle X component Y component
4.5N 30o 3.89N 2.25N
7N 210o -6.06N -3.5N
6N 150o -5.19N 3.0N
--------------- ------------ Rx=-7.36N Ry=1.7N
cosxA A 4.5 @30oA N
7 @ 210oB N
6 @150oC N
sinyA A cosxB B cosxC C
sinyB B sinyC C
R=7.55N Angle =-13o+180o = 167o
• Analytically and Graphically add the following vector sets.
• v1 17m/s @ 300• v2 24m/s @ 170• v3 24m/s @ 55o
• v4 19m/s @ 20o
Adding Vectors WS 17
Practice Problem WS6a #117 @300om
sA
24 @170omsB
24 @55omsC
R=22.7m/s Angle = 43.4o
19 @ 20omsD
Magnitude Angle Rx Ry
17 300 8.5 -14.7
24 170 -23.6 4.17
24 55 13.76 19.65
19 20 17.85 6.49
-------------- ------------ 16.51 15.62
Adding “-” Vectors
C = A + B
D = A - B
D = A + (- B)
A
B
C -B
D
Add “negative” vectors by keeping the same magnitude but adding 180 degrees to the direction of the original vector.
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) Rotate it
b) Move it
c) Lengthen it
d) 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 10m/s @ 45– V2 15m/s @ 135
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 ?
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
cosxA A 7 @ 45oA N
8 @180oB N
6 @ 270oC N
Ay
Ax
=45o
A=7N
sinyA A
cosxB B
cosxC C
sinyB B
sinyC C
B=8
C=
6
Add the x components togetherAdd the y components togetherCompute the Resultant
Adding Vectors Analytically
Magnitude Angle X Y
7N 45o 4.95N 4.95N
8N 180o -8N 0N
6N 270o 0N -6N
--------------- ------------ Rx=-3.05N Ry=-1.05N
cosxA A 7 @ 45oA N
8 @180oB N
6 @ 270oC N
sinyA A cosxB B cosxC C
sinyB B sinyC C
R=3.22N Angle =19o+180o = 199o
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