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8/8/2019 Lecture 0- REVIEW- Scalar, Vector
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Scalar & Vector
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Scalar & Vector
Vector Analysis and
Coordinate System
LESSON 1
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Define scalar and vector quantities, unit vector inCartesian coordinate.
Vector addition operation and their rules and
visualize resultant vector graphically by applyinga) commutative
b) associative
c) distributive and rules
Objectives
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A quantity that has both magnitude and direction.
Examples: velocity, acceleration, force, and momentum.
Vector notation : or A or
A vector can be represented by an arrow ;
Scalar quantity
A quantity that has magnitudebut no direction.
Examples : mass, work, speed, energy, and density.
Vector quantity
Ar
vector direction
Ar a
%
magnitude of A AA = orr r
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Pr
Qr
P Q and point in theP Q same dire tionc !r rrr
Equality of two vectors
Two vectors are equal if they have equal lengths and
point in thesame direction.
Pr
Negative of a vector
The negative of a vector is vector having thesamelengthbut opposite direction.
-Pr
Multiplying a vector by a scalar
m P = mP ; = mPmagnitude
direct same directionion of P
v
|
r r r
r
positive scalar quantity :
negative scalar quantity :
(-m) P -mP ; mPmagnitude
directi opposite directionon of P
v
|
r r r
r
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A Er r
equal vectors :
negative of a vector : D -A - Err r
parallel vectors :
anti parallel vectors :
A ; E ; F ; Gr rr r
A and D ; A and H ; F and H ; D and E ;E and H ; F and D ; D and G ; G and H
r rr r r r r r
r rr r r r r r
F A2rr 1
2
AG A
2
rr r
Example :
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Quiz :
1. Define scalar quantity
2. Define vector quantity
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Vector Addition & Subtraction
Vector addition
Vector addition obeys commutative, associative and distributive
laws
A resultant vector is a single vector which produces the same
effect ( in both magnitude and direction) as the vector sum of
two or more vectors.
2 methods of vector addition :
graphical method - head to tail / tip to tail
-parallelogram
calculation /algebraically (component method)
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Ar
AT
BT
B r C! r
CT
Additional graphical : head to tail
Example 1
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AT B
T
CT
DT
DCBARTTTTT
!
Additional graphical : head to tail
Example 2
eometric construction for
summing four vectors.
is the resultant vectorRr
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AT
A
T
BT
BT
CT
C=A+Br r r
Additional graphical : parallelogram
Example
The resultant vector , is the
diagonal of the parallelogram.
Cr
- The resultant of two vectors acting at any angle may be represented
by the diagonal of a parallelogram.
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AT
A
T
BT
BT
CT
Commutative law :
Example
Vector addition is commutative
CABBA
TTTTT
!!
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Associative law :
Vector addition is associative
Example
)CB(ATTT
CBTT
AT
BT
CT
C)BA(TTT
BATT
AT
BT
CT
)CB(ATTT
C)BA(TTT
=
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Distributive law :
AT B
T
BATT
AmT
BmT
)BAm(TT
BmAm)BAm(TTTT
!Vector addition is distributive
Example
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Vector subtraction :
A - B = A (-B) = Cr r rr r
AT
BT
CT
B-T
The vector is equal in magnitude to vector
and points in the opposite direction.
-Br
Br
To subtract from , apply the rule of vector
addition to the combination of and
Br
Ar
-Br
Ar
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1.4 Scalar & Vector
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Vector addition & subtraction
Example 1
Ar
B
r
Draw the vectors :
a ) A B
b ) A - B
c ) 2 A B
r r
r r
r r
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Vector addition & subtraction
Example 1
Ar
B
r
Draw the vectors :
a ) A B
b ) A - B
c ) 2 A B
r r
r r
r r
Solution
A
rBr
A+Br r
a) b)
Ar
-B
r
A - Br r
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c) 2A+ B
r r
2Ar
Br
2A+ Br r
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Components of a vector & Unit Vectors
Components of a vector :
)U
AT
y
x
can be resolved into its components that areAT
A vectorperpendicular to each other.
i) In 2 D
Axr
Ayr
VectorA is resolved into a-component -componentndx yr
(vector components)
A = A +A x yr r r
A = = A cos or cosx xA AU Ur r
A = A sin or siny yA AU U!
r r
1tany
x
A
AU !
Direction :Magnitude :
2 2
x y A A A! r
(scalar
components)
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A =r
Ax
Ay)E
F
K
AT
y
x
z
ii) In 3 D
Vector A can be resolved into 3 c ,omponents : a compond nents x y z
r
Axr
Ayr
Az
r
A = = A cosx xA Er r
A = A cosy yA F!r r
A = A cosz zA K!r r
Az
Magnitudes :
Axr
+Ayr
+Azr (vector
components)
(scalar
components)
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Ar
xAr
yA
r
zAr
x
y
z
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REMEMBER !!!
x
y
Ax positive
Ax positive
Ay positive
Ay positive
Ax negative
Ax negative
Ay negativeAy negative
The signs of the components of a vector(eg: vector )
depend on the quadrant in which the vector is located :
Ar
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E
Addition of vectors using components ( Cartesian coordinates)
x
y
Ar
Br
Cr
F
K
A = + = cos sinx y A A A AE Er
B = + = cos sinx y B B B BF Fr
C = + = cos sinx yC C C C K Kr
Vectors x-component y-component
A
r
Br
Cr
Let R is the resultant vector,r
cos cos cos x x x xR A B C A B C E F K! ! sin sin sin y y y y A B C A B C E K! !
2 2agnitude, x yR R!
r
-1
irection , = tan
y
x
R
RU
cosA E sinA EcosB F sinB F
cosC K sinC K
Ry
Rx
Rr
U x
y
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Forces,F x-component y-component
37o
Addition of vectors using components ( Cartesan coordinates)
x
y
45o
Let R is the resultant vector,r
(80 71 95 150) = -94x xR F! !
2 2agnitude, R ( 94) (71) 118F! ! !r
-1 o71Direction , = tan 3794
U !
Example 1: our coplanar forces act on a body at point O. ind their resultant.
100 N
110 N
160 N
20o
O 80 N
30o
(0 71 55 55)N = 71 Ny yR F! !
71 N
-94 N
Rr
143o
(or, 143
o
from positivex-axis )
Solution :
80 N 80 cos 0o =80 N
100 N
80 sin 0o = 0
100 cos 45o = 71 N 100 sin 45o = 71 N
110 N -110 cos 30o = -95 N 110 sin 30o =55 N
160 N -160 cos 20o = -150 N -160 sin 20o = -55 N
x
y
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A dimensionless vector
ave magnitude of 1, with no units.
Show direction
Examples :
Unit Vectors :
ij
k
ij
k
: unit vector in the +ve x- direction: unit vector in the +ve y-direction
: unit vector in the +ve z-direction
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A ix yA A! r
)U
A
T
y
x
Components of a vector in the form of unit vector :
i) In 2 Dii) In 3 D
A i k x y z A A A! r
AT
y
x
z
jyA
jyA
ix
A
ixAkzA
Example :
A 2i 3j! r Example :
A 3i 4 j+ 2k! r
2
2
0
0
3
3
4
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Example 1:
A (6i 3j k) and B (4i 5 j 8k)! ! r riven two vectors
Solution :
C = A+ B
(6i 3j k) (4i -5j+8k)
(6 4)i +(3-5)j+(-1+8)k
10i - 2j+ 7k
!
! -
!
r r r
Addition of vectors using components ( unit vectors)
ind C A B and its magnitude.! r r r
2 2 2agnitude, C 10 ( 2) 7 12.4! !
r
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Example 2:
P =(2i + 3j) and Q (3i - 2j + 3k)!rriven two vectors
Solution :
R= 2P - Q =2(2i + 3j + 0k) - (3i - 2j + 3k)
= (4i + 6j + 0k) - (3i - 2j + 3k)
= (4 - 3)i + (6 + 2)j + (0 - 3)k
= i + 8j - 3k
-
-
rr r
Find R 2P Q and its magnitude.! rr r
2 2 2agnitude, R 1 8 ( 3) 8.6! !r
Addition of vectors using components ( unit vectors)
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Example 3:
A (6i 3j) and B (4i 5 j)! ! r riven two vectors
Solution :
C = A+ B
(6i 3j) (4i -5j)
(6 4)i +(3-5)j
10i - 2j
!
! -
!
r r r
Addition of vectors using components ( unit vectors)
ind C A B and its magnitude.! r r r
2 2Magnitude, C 10 ( 2) 10.2! !r
Direction ;
1
1
0
tan
( 2) = tan
10
= -11.3
y
x
C
CU
!
x
y
-2
10
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- graphical
A = i + j+ k x y z A A Ar
x y A = i + jA Ar
Conclusion
head to tail
parallelogram
Scalar quantity has magnitude
Vector quantity has magnitude and direction
Unit vector magnitude 1 ; has no unit ; shows direction
Component of a vector in 2-D :
Component of a vector in 3-D :
Vector addition :
- by calculation ( component method)