vectors - universe of ali ovgun · january 21, 2015 properties of vectors q equality of two vectors...
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Physics 101Lecture 1Vectors
Assist. Prof. Dr. Ali ÖVGÜN
EMU Physics Department
www.aovgun.com
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Coordinate SystemsqCartesian
coordinate systemqPolar coordinate
system
January 21, 2015
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qFrom Cartesian to Polar coordinate system
qFrom Polar to Cartesian system
January 21, 2015
Answer: (4.30 m, 216)
Direction:
Magnitude
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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!
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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!
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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!
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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!
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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.
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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!!!
+=
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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,
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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
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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 +=
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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
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Example 2 :
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Example 3 :
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January 21, 2015
Multiplying Vectors
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January 21, 2015
Scalar Product
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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
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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× = − ×! !! !
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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)( −+−+−=×
!!
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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× =! !
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January 21, 2015
Example 6:
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Example 7:
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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
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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
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January 21, 2015
Problem 5
Problem 6
Problem 7
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Problem 8
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