Physics 101Lecture 1Vectors

Assist. Prof. Dr. Ali ÖVGÜN

EMU Physics Department

www.aovgun.com

Coordinate SystemsqCartesian

coordinate systemqPolar coordinate

system

January 21, 2015

qFrom Cartesian to Polar coordinate system

qFrom Polar to Cartesian system

January 21, 2015

Answer: (4.30 m, 216)

Direction:

Magnitude

January 21, 2015

Vector vs. Scalar Review

q All physical quantities encountered in this text will be either a scalar or a vector

q A vector quantity has both magnitude (number value + unit) and direction

q A scalar is completely specified by only a magnitude (number value + unit)

A library is located 0.5 mi from you. Can you point where exactly it is?

You also need to know the direction in which you should walk to the library!

January 21, 2015

Vector and Scalar Quantities

q Vectorsn Displacementn Velocity (magnitude and

direction!)n Accelerationn Forcen Momentumn Weight

q Scalars:n Distancen Speed (magnitude of

velocity)n Temperaturen Massn Energyn Time

To describe a vector we need more information than to describe a scalar! Therefore vectors are more complex!

January 21, 2015

Important Notationq To describe vectors we will use:

n The bold font: Vector A is An Or an arrow above the vector: n In the pictures, we will always show

vectors as arrowsn Arrows point the directionn To describe the magnitude of a

vector we will use absolute value sign: or just A,

n Magnitude is always positive, the magnitude of a vector is equal to the length of a vector.

A!

A!

January 21, 2015

Properties of Vectorsq Equality of Two Vectors

n Two vectors are equal if they have the same magnitude and the same direction

q Movement of vectors in a diagramn Any vector can be moved parallel to

itself without being affected

A!

q Negative Vectorsn Two vectors are negative if they have the same

magnitude but are 180° apart (opposite directions)

( ); 0= − + − =A B A A! ! ! !

B!

January 21, 2015

Describing Vectors AlgebraicallyVectors: Described by the number, units and direction!

Vectors: Can be described by their magnitude and direction. For example: Your displacement is 1.5 m at an angle of 250.Can be described by components? For example: your displacement is 1.36 m in the positive x direction and 0.634 min the positive y direction.

January 21, 2015

Components of a Vectorq The x-component of a vector is

the projection along the x-axis

q The y-component of a vector is the projection along the y-axis

q Then,

x y= +A A A! ! !θ

cos xAA

θ = cosxA A θ=

sin yAA

θ = sinyA A θ=

yx AAA!!!

+=

January 21, 2015

More About Componentsq The components are the legs of

the right triangle whose hypotenuse is A

2 2 1tan yx y

x

AA A A and

Aθ − ⎛ ⎞

= + = ⎜ ⎟⎝ ⎠

( ) ( )

( )⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛==

+=

⎩⎨⎧

==

−

x

y

x

y

yx

y

x

AA

AA

AAA

AAAA

1

22

tanor tan

)sin()cos(

θθ

θθ

!

θ Or,

January 21, 2015

Unit Vectorsq Components of a vector are vectors

q Unit vectors i-hat, j-hat, k-hat

q Unit vectors used to specify directionq Unit vectors have a magnitude of 1q Then

x y= +A A A! ! !

θ

zk→ˆxi →ˆ yj→ˆ

y

x

z

ij

k

yx AAA!!!

+=

jAiAA yxˆˆ+=

!

Magnitude + Sign Unit vector

January 21, 2015

Adding Vectors Algebraicallyq Consider two vectors

q Then

q Ifq so x y= +A A A

! ! ! jBAiBABAC yyxxˆ)(ˆ)( +++=+=

!!!jBAiBA

jBiBjAiABA

yyxx

yxyx

ˆ)(ˆ)(

)ˆˆ()ˆˆ(

+++=

+++=+!!

jBiBB yxˆˆ+=

!jAiAA yxˆˆ+=

!

xxx BAC += yyy BAC +=

January 21, 2015

Example 1: Operations with Vectorsq Vector A is described algebraically as (-3, 5), while

vector B is (4, -2). Find the value of magnitude anddirection of the sum (C) of the vectors A and B.

jijiBAC ˆ3ˆ1ˆ)25(ˆ)43( +=−++−=+=!!!

jiB ˆ2ˆ4 −=!

jiA ˆ5ˆ3 +−=!

1=xC 3=yC

16.3)31()( 2/1222/122 =+=+= yx CCC!56.713tan)(tan 11 === −−

x

y

CC

θ

January 21, 2015

Example 2 :

January 21, 2015

Example 3 :

January 21, 2015

Multiplying Vectors

January 21, 2015

Scalar Product

September 30, 2018

Cross Product

q The cross product of two vectors says something about how perpendicular they are.

q Magnitude:

n θ is smaller angle between the vectorsn Cross product of any parallel vectors = zeron Cross product is maximum for perpendicular

vectorsn Cross products of Cartesian unit vectors:

θsinABBAC =×=→ !!

BAC!!

×=→

A!

B!

θ

θsinA!

θsinB!

0ˆˆ ;0ˆˆ ;0ˆˆ

ˆˆˆ ;ˆˆˆ ;ˆˆˆ

=×=×=×

=×−=×=×

kkjjii

ikjjkikji

y

xz

ij

k

i

kj

September 30, 2018

Cross Productq Direction: C perpendicular to

both A and B (right-hand rule)n Place A and B tail to tailn Right hand, not left handn Four fingers are pointed along

the first vector An “sweep” from first vector A

into second vector B through the smaller angle between them

n Your outstretched thumb points the direction of C

q First practice? ABBA!!!!

×=×

? ABBA!!!!

×=×

A B B A× = − ×! !! !

September 30, 2018

More about Cross Productq The quantity ABsinθ is the area of the

parallelogram formed by A and Bq The direction of C is perpendicular to

the plane formed by A and Bq Cross product is not commutative

q The distributive law

q The derivative of cross productobeys the chain rule

q Calculate cross product

( )dtBdAB

dtAdBA

dtd

!!!

!!!

×+×=×

CABACBA!!!!!!!

×+×=+× )(

A B B A× = − ×! !! !

kBABAjBABAiBABABA xyyxzxxzyzzyˆ)(ˆ)(ˆ)( −+−+−=×

!!

September 30, 2018

Examples of Cross Products

mjirNjiF )ˆ5ˆ4( )ˆ3ˆ2( +=+= !!

?A B×! !

Solution: i

kj

jiBjiA ˆ2ˆ ˆ3ˆ2 +−=+=!!

kkkijji

jjijjiii

jijiBA

ˆ7ˆ3ˆ40ˆˆ3ˆˆ40

ˆ2ˆ3)ˆ(ˆ3ˆ2ˆ2)ˆ(ˆ2

)ˆ2ˆ()ˆ3ˆ2(

=+=+×−×+=

×+−×+×+−×=

+−×+=×!!

Ex.5: Calculate 𝒓×𝑭given a force and its location

ˆ ˆ ˆ ˆ(4 5 ) (2 3 )ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ4 2 4 3 5 2 5 3

ˆ ˆ ˆˆ ˆ ˆ ˆ0 4 3 5 2 0 12 10 2 (Nm)

r F i j i j

i i i j j i j j

i j j i k k k

× = + × +

= × + × + × + ×

= + × + × + = − =

!!Solution:

Ex. 4: Find: Where:

ˆˆ ˆ

4 5 02 3 0

i j kA B× =! !

January 21, 2015

Example 6:

January 21, 2015

Example 7:

January 21, 2015

Summaryq Polar coordinates of vector A (A, θ)q Cartesian coordinates (Ax, Ay)q Relations between them:q Beware of tan 180-degree ambiguityq Unit vectors:q Addition of vectors:

q Scalar multiplication of a vector:

( ) ( )

( )

22

1

cos( )

sin( )

tan or tan

x

y

x y

y y

x x

A AA A

A A A

A AA A

θθ

θ θ −

=⎧⎪⎨ =⎪⎩⎧ = +⎪⎪⎨ ⎛ ⎞⎪ = = ⎜ ⎟⎪ ⎝ ⎠⎩

ˆˆ ˆx y zA i A j A k= + +A

jBAiBABAC yyxxˆ)(ˆ)( +++=+=

!!!

xxx BAC += yyy BAC +=

ˆ ˆx ya aA i aA j= +A

January 21, 2015

Problem 1:

A particle undergoes three consecutive displacements: !! = 15! + 30! + 12! !, !! = 23! − 14! − 5! ! and !! = −13! + 15! !. Find

the components of the resultant displacement and its magnitude.

Ans: ! = 25! + 31! + 7! ! and ! = 40!

Problem 2:

The polar coordinates of a point are ! = 5.5! and ! = 240°. What are the Cartesian coordinates of this point?

Sln: ! = !"#$% = 5.5! !"#240° = 5.5! −0.5 = −2.75!

! = !"#$% = 5.5! sin 240° = 5.5! −0.866 = −4.76!

Problem 3:

If ! = (3! − 4! + 4!) ve ! = 2! + 4! − 8!);

(a) Express ! in unit vector notation, ! = (2! − !!!).

(b) Find the magnitude and direction of !.

(-2.75,-4.76)m

January 21, 2015

January 21, 2015

January 21, 2015

January 21, 2015

January 21, 2015

January 21, 2015

January 21, 2015

Problem 5

Problem 6

Problem 7

January 21, 2015

Problem 8

January 21, 2015

January 21, 2015

January 21, 2015