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Ch03:VectorsVector and Scalar quantitiesAdding vectors geometrically and vectors propertiesComponents of vectorsUnit VectorsAdding vectors by componentsMultiplying vectors (scalar and vectoproduct)

3.1: What is physics?In physics we usually deals with many quantities that have both size and direction (vector quantities) like displacement, velocity, acceleration, etc….Not like in chapter two, vector quantities usually exist in more than one dimension.This need a special mathematical language (language of vectors) used by scientists and engineersIn this chapter we will focus on basic language of vectors

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3.2: Scalars and VectorsScalars: quantities only have a single value (only magnitude) and an appropriate unit.

Distance (5 m), time (4 sec.), mass (5 gr.), temperature (15°C),……

Vectors: quantities that have both magnitude and direction coupled with a unit.

Displacement (5 ft. to left), velocity (10 mph, due north), acceleration (9.8 m/s2, downward),….;

A

B

The curved line: distance; the path takenThe red arrow: Displacement vector

Vectors are represented by arrows.

The head of the arrow signifies direction; Tip points away from the starting point in the direction of the vector to the ending point.

Denoted by or

the length of the arrow signifies the magnitude

and is denoted by or

Vectors are equal if they have the same magnitude and direction.

ar a

ar a

3.2: Scalars and Vectors

ar

3

Draw vector

Draw vector starting from the tip of

Draw the resultant vector

from the beginning of to the tip of

3.3: Adding vectors geometricallyar

basrrr

+=ar b

r

br

ar

Same is followed for more than two vectors

Cumulative law ( تبديلي):

Associative law (تجميعي):

3.3: Adding vectors geometrically: adding rules

for any order of adding the vectors same result.

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has the same magnitude as but is in the opposite direction.

To subtract , just add the negative of

3.3: Adding vectors geometrically: negative of a vector and vector subtraction

br

− br

0)( =−+ bbrr

ab rrfrom ab rr

to

In solving equations, we must change the sign of the vector when moving it from one side to other

3.3: Adding vectors geometrically: multiplying a vector by a scalar

If the scalar is positive, the direction of the vector does not change but its magnitude is multiplied by the scalar value.If it is negative, its direction is reversed and its magnitude multiplied.

ar2ar ar2−

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3.3: Adding vectors geometrically: Example: adding vectors in drawing

You have the vectors shown below, find the magnitude of the vector sum

to make the sum shown by drawing, we need to use a ruler and a Protractor

Using ruler to measure the magnitude of vector result (note given scale)

Components of a vector projections along coordinate axis (x, y, z)3.4: Components of vectors

a

yaaa

xaa

y

x

r

r

r

r

ofcomponent -y

axisalongvectorofprojection: ofcomponent -x

axisalongvectorofprojection:

⇒

−⇒

−

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From the right angle triangle, we can find:

3.4: Components of vectors

x

y

x

y

aaaaaa

=

=

=

θ

θ

θ

tan

cos

sin

Where θ is the angle between the vector and the +ve x-axis

A plane flies 215 km 22° east of north. How far due east and due north the plane flies?

3.4: Components of vectors: Example

Sol’n: let the displacement be vector

θ from x-axis (east) is 90-22=68°

due east

due north

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3.5: Unit Vectors

Unit vector isA dimensionless vector with a magnitude of 1.

Used only to specify direction.

The represent unit vectors in the x, y and z directions respectively.

1ˆˆˆ === kjikji ˆ,ˆ,ˆ

To identify the direction for vector components, we canuse what is called Unit vector.

3.5: Unit Vectors

²²

ˆˆˆ

yx

yx

AAjAiA

AAA

+

+== r

r

Unit vector can be introduced in any direction like unit vector  in direction of A , where

vector then can berepresented with unit vectors as

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3.6: Adding vectors by components

jaiaa yxˆˆ +=

r jbibb yxˆˆ +=

r

( ) ( )( ) ( )

jrirr

jbaibar

jbibjaiabar

yx

yyxx

yxyx

ˆˆ

ˆˆ

ˆˆˆˆ

+=

+++=

+++=+=

r

r

rrr

( ) ( )xx

yy

x

yyyxxyx ba

barr

θ and babarrr+

+==+++=+= tan)( 2222

Adding vectors is done by adding each component type (x components together and y components together) for the vectors together. Assume the two vectors

yyyxxx barandbar +=+=Hence,

a: Find magnitude and direction ofb: find unit vector in the direction of

if mjibmjia )ˆ4ˆ2( )ˆ2ˆ2( −=+=rr

mjir

jira

)ˆ2ˆ4(

ˆ)42(ˆ)22( )

−=

−++=r

r

°−=−

===>

===>=

==−+=

−

−

2742tan

tantan

5.420²2²4

1

1

θ

θθx

y

x

y

RR

RR

mr

jijibbbb ˆ

204ˆ

202

²4²2

ˆ4ˆ2ˆ ) −+=

−+−

==r

barrrr

+=br

3.6: Adding vectors by components: Example

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3.6: Adding vectors by components: Example

Given the three vectors below, find the resultant cbar rrrr++=

jrirr yxˆˆ +=

r

Rearrange vectors as below

Airplane flies from the originto city A, located 175 km in a direction 30.0° north of east.Next, it flies 153 km 20.0° westof north to city B. Finally, itflies 195 km due west to city C. Find the location of city Crelative to the origin.

3.6: Adding vectors by components: Example

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ExampleSolution: we have 3 vectors as shown; Theresultant

°=−

=

=+−=+=

+−=

+++++=+=

−

8.1123.95

232tan

251)²232()²3.95(²²

ˆ232ˆ3.95

ˆ)(ˆ)(

1

θ

kmRRR

jiR

jcbaicbajRiRR

yx

yyyxxxyxr

r

jijcicjcicc

jijbibjbibb

jijaiajaiaa

yx

yx

yx

ˆ0ˆ195ˆ180sinˆ180cosˆˆ

ˆ144ˆ3.52ˆ110sinˆ110cosˆˆ

ˆ88ˆ152ˆ30sinˆ30cosˆˆ

+−=°+°=+=

+−=°+°=+=

+=°+°=+=

r

r

r

candba rrr ,,cbaR rrrr

++=

3.7: vectors and laws of physicsVectors do not depend on the location of the origin or on the

orientation of the axes; they are all independent of the choice of coordinate system no change in laws of physics

If coordinate system is rotated by an angle φ components changes to without change in the vector. '' and yx aa

and

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φcosabba =⋅rr

Dot product: is the product of a and the projection of vector on vector

3.8 Multiplying vectors: The Scalar or Dot Product of Two Vectors:

φcosbabarrrr

=⋅

abba =⋅rr

br

0. =barr

abba −=rr.

(scalar quantity)

ar

If =0°

If =90°

If =180°

For unit vectors

Dot product characteristics

1) Commutative

2) Distributive in multiplication

0ˆˆˆˆˆˆ =⋅=⋅=⋅ kjkiji

abba rrrr .=⋅( ) cabacba rrrrrrr

⋅+⋅=+⋅

kajaiaa zyxˆˆˆ ++=

r

kbjbibb zyxˆˆˆ ++=

rzzyyxx babababa ++=

rr.

2. aaaaaaaaa zzyyxx =++=rr

1ˆˆˆˆˆˆ =⋅=⋅=⋅ kkjjii ( ⁄⁄ )

(┴)For vectors represented by components

3.8 The Scalar or Dot Product of Two Vectors: properties of scalar product

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Example

ba

barr

rr

and between angle b)

a)

φ

⋅

abbaabba

bababa yyxxrrrr

rr

⋅=⇒=⋅

=+−=+=⋅

−1coscos b)

462 a)

:solution

φφ5²2²1

13²3²2

=+−==

=+==

bb

aar

r

findj.i.b and j.i.aIf ˆ02ˆ01ˆ03ˆ02 +−=+=rr

°==⇒ − 2.60654cos 1φ

bacrrr

×=

φsinabc =

For two vectors and , the vector product is a third vector so that

(if θ = 0° c = 0, if θ = 90° c = ab)

br

ar

The magnitude of the vector iscrcr

direction is perpendicular (┴) to both vectors and cr

br

ar

You can use the right-hand rule to find the direction of . We begin with first vector in the cross product and 0<θ<180°

cr

3.8 Multiplying vectors: The vector or cross Product of Two Vectors:

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abba rrrr×−=×

0=×aa rr

( ) cabacba rrrrrrr×+×=+×

( )dtbdab

dtadba

dtd

rrrrrr×+×=×

jkiik

ijkkj

kijji

ˆˆˆˆˆ

ˆˆˆˆˆ

ˆˆˆˆˆ

=×−=×

=×−=×

=×−=×

k)bab(aj)bab(ai)bab(abbbaaakji

ba xyyxxzzxyzzy

zyx

zyxˆˆˆ

ˆˆˆ

−+−−−==×rr

For vectors represented by components

kajaiaa zyxˆˆˆ ++=

r

kbjbibb zyxˆˆˆ ++=

r

Unit vectors cross product

3.8 The vector or cross Product of Two Vectors: properties

Example

302043

ˆˆˆ

−−=×

kjibarr

kjibaab ˆ8ˆ9ˆ12 ++=×−=×rrrr

?ab ? ba

findkib and jiaIf

=×=×

+−=−=rrrr

rr ˆ3ˆ2ˆ4ˆ3

kji

kjiˆ8ˆ9ˆ12

))2(4)0(3(ˆ))2(0)3(3(ˆ))0(0)3(4(ˆ

−−−=

−−−+−−−−−=

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Example:

axiszandBbetweenangleh

axiszandAbetweenangleg

Aofdirectiontheinvectorunitf

e

BandAbetweenangled

c

b

a

k

−

−

×

⋅

−

+

++−=

+=

r

r

r

rr

rr

rr

rr

rr

r

r

)

)

)

)

)

)

2)

)

find ˆ2ˆ4ˆ and ˆ3ˆ2 If

BA

BA

BA

BA

jiB

jiA