lecture 0- review- scalar, vector

8/8/2019 Lecture 0- REVIEW- Scalar, Vector

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Scalar & Vector

1

Scalar & Vector

Vector Analysis and

Coordinate System

LESSON 1


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Scalar & Vector

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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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Scalar & Vector

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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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Scalar & Vector

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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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Scalar & Vector

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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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Scalar & Vector

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Quiz :

1. Define scalar quantity

2. Define vector quantity


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Scalar & Vector

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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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Scalar & Vector

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Ar

AT

BT

B r C! r

CT

Additional graphical : head to tail

Example 1


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Scalar & Vector

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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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Scalar & Vector

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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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Scalar & Vector

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AT

A

T

BT

BT

CT

Commutative law :

Example

Vector addition is commutative

CABBA

TTTTT

!!


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Scalar & Vector

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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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Scalar & Vector

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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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Scalar & Vector

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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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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

Solution

A

rBr

A+Br r

a) b)

Ar

-B

r

A - Br r


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Scalar & Vector

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c) 2A+ B

r r

2Ar

Br

2A+ Br r


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Scalar & Vector

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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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Scalar & Vector

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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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Scalar & Vector

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Scalar & Vector

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Ar

xAr

yA

r

zAr

x

y

z


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Scalar & Vector

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Scalar & Vector

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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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Scalar & Vector

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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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Scalar & Vector

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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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Scalar & Vector

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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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Scalar & Vector

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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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Scalar & Vector

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Scalar & Vector

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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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Scalar & Vector

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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



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Scalar & Vector

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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


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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Scalar & Vector

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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)

lecture 0- review- scalar, vector

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