Lec1 Vectors

8/10/2019 Lec1 Vectors

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Lecture 1

VECTORS

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Lecture 1

VECTORS

1 Physical Quantities

2 Operations with Vectors

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Lecture 1

VECTORS



3 Cartesian Coordinates

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Lecture 1

VECTORS



3 Cartesian Coordinates

4 Time rate of change of a vector

5 Advanced Topics

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Physical Quantities

Scalars :

Vectors :

Vectors Physical Quantities 2/9

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Physical Quantities

Scalars : described by a single number (magnitude)

Vectors : direction & magnitude


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Physical Quantities


Mass, Temperature, SpeedVectors : direction & magnitude

Velocity, Acceleration, Angular Momentum, Electric, Magnetic Field


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Physical Quantities


Mass, Temperature, Speed



Vectors are


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Physical Quantities





Vectors are

Arrows in Space


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Physical Quantities





Vectors are

Arrows in Spacedenoted (always!) by a symbol with anarrow sign on top:

−→A


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Physical Quantities





Vectors are


−→A

Magnitude of−→A denoted as |

−→A | or

simply A


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Physical Quantities





Vectors are


−→A


−→A | or

simply A

Tensors : Generalization of vectors:


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Physical Quantities





Vectors are


−→A


−→A | or

simply A

Tensors : Generalization of vectors: products of vectors


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Physical Quantities





Vectors are


−→A


−→A | or

simply A

Tensors : Generalization of vectors: products of vectors

Moment of Inertia, Stress tensor, permeability, energy-momentum


h

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Operations with Vectors

Vectors Operations with Vectors 3/9

O i i h V

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Addition:−→

A +−→

B =−→

C


O i i h V

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Addition:−→

A +−→

B =−→

C

Multiplication by scalar:

a ×−→A =

−→aA


O ti ith V t

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Addition:−→

A +−→

B =−→

C


a ×−→A =

−→aA

Scalar Product:−→A ·

−→B = AB cos θ


O ti ith V t

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Addition:−→

A +−→

B =−→

C


a ×−→A =

−→aA

Scalar Product:−→A ·

−→B = AB cos θ

Cross Product −→A ×

−→B =

−→C


Vectors: Component form

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Description relative to a set of coordinates in 3d Space.

$A_y$

$A_x$

$A_z$

$x$

$z$

$y$

−→A

Vectors Cartesian Coordinates 4/9

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Cartesian coordinates

$A_y$

$A_x$

$A_z$

y

z

−→A

ˆ

k

x

ˆ

i

ˆ

j

Orthogonal unit vectors,right-handed systemi , j , k



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$A_y$

$A_x$

$A_z$

y

z

−→A

ˆ

k

x

ˆ

i

ˆ

j


î ·

î =

ˆ j ·

ˆ j =

ˆk ·

ˆk = 1



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$A_y$

$A_x$

$A_z$

y

z

−→A

ˆ

k

x

ˆ

i

ˆ

j


î ·

î =

ˆ j ·

ˆ j =

ˆk ·

ˆk = 1

i · j = j · k = k · i = 0



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$A_y$

$A_x$

$A_z$

y

z

−→A

ˆ

k

x

ˆ

i

ˆ

j


î ·

î =

ˆ j ·

ˆ j =

ˆk ·

ˆk = 1

i · j = j · k = k · i = 0

i × j = k ; j × k = i ;k × i = j .



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$A_y$

$A_x$

$A_z$

y

z

−→A

ˆ

k

x

ˆ

i

ˆ

j

Decomposition of Vectorinto components:



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y

z

−→A

Ay

Az

ˆ

k

x

Ax

ˆ

i

ˆ

j

Decomposition of Vectorinto components:Projections along theaxes



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y

z

−→A

Ay

Az

ˆ

k

x

Ax

ˆ

i

ˆ

j


−→A = Ax i + A y j + A z k



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p



y

z

−→A

Ay

Az

ˆ

k

x

Ax

ˆ

i

ˆ

j


−→A = Ax i + A y j + A z k

i , j , k are constantvectors.


Operations with Vectors: Component form

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p p



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p p

Magnitude of−→A : A = A2

x + A 2y + A 2

z



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p p


x + A 2y + A 2

z

Vector Addition:−→C =

−→A +

−→B

= ( Ax + B x ) i + ( Ay + B y ) j + ( Az + B z )k

= C x i + C y j + C z k



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x + A 2y + A 2

z

Vector Addition:−→C =

−→A +

−→B

= ( Ax + B x ) i + ( Ay + B y ) j + ( Az + B z )k

= C x i + C y j + C z k

Scalar Product:

−→

A ·−→

B = Ax B x + A y B y + A z B z



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Vector Product:

−→C =

−→A ×

−→B =

ˆ

i ˆ

j ˆ

kAx Ay Az

B x By B z

= ( Ay B z − A z B y ) i + ( Az B x − A x B z ) j + ( Ax B y − A y B x )k



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Vector Product:

−→C =

−→A ×

−→B =

ˆ

i ˆ

j ˆ

kAx Ay Az

B x By B z


Direction of vector product: right hand screw rule



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Vector Product:

−→C =

−→A ×

−→B =

ˆ

i ˆ

j ˆ

kAx Ay Az

B x By B z


Direction of vector product: right hand screw rule

−→A ×

−→B

−→A

−→B

θ


Time Rate of Change of a Vector

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O

Vectors Time rate of change of a vector 7/9


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O



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O

trajectory



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O

r(t)

trajectory



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O

r(t)r (t + ∆ t)

trajectory



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O

r(t)r (t + ∆ t)

∆ r

trajectory



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O

r(t)r (t + ∆ t)

∆ r

trajectory lim∆ t → 0

−→∆

r∆ t =



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O

r(t)

r (t + ∆ t)

∆ r


−→∆

r∆ t = d−→

rdt =



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O

r(t)

r (t + ∆ t)

∆ r


−→∆

r∆ t = d−→

rdt = dxdt i + dydt j + dzdt k



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O

r(t)

r (t + ∆ t)

∆ r


−→∆

r∆ t = d−→


d−→A

dt =



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O

r(t)

r (t + ∆ t)

∆ r


−→∆

r∆ t = d−→


d−→A

dt =

dAx

dti +

dAy

dt j +

dA z

dtk


Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes

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Transformation of Vectors under Rotation of Coordinate Axes

x

y

Vectors Advanced Topics 8/9


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x

y

A



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x

y

AAy

Ax



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y ′

x ′

x

y

AAy

Ax

θ



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y ′

x ′

x

y

AAy

Ax

A ′

y A ′

xθ



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y ′

x ′

x

y

AAy

Ax

A ′

y A ′

xθ

−→A = Ax i + A y

ˆ j

= Ax i + A y j



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y ′

x ′

x

y

AAy

Ax

A ′

y A ′

xθ

−→A = Ax i + A y

ˆ j

= Ax i + A y j

Ax = Ax cos θ + A y sin θAy = − Ax sin θ + A y cos θ



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y ′

x ′

x

y

AAy

Ax

A ′

y A ′

xθ

−→A = Ax i + A y

ˆ j

= Ax i + A y j


AxAy



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y ′

x ′

x

y

AAy

Ax

A ′

y A ′

xθ

−→A = Ax i + A y

ˆ j

= Ax i + A y j


AxAy

= cosθ sin θ− sin θ cosθ



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y ′

x ′

x

y

AAy

Ax

A ′

y A ′

xθ

−→A = Ax i + A y

ˆ j

= Ax i + A y j


AxAy

= cosθ sin θ− sin θ cosθ AxAy



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Ax

Ay



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Ax

Ay =

R xx Rxy

R yx R yy



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Ax

Ay =

R xx Rxy

R yx R yy

Ax

Ay



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Ax

Ay =

R xx Rxy

R yx R yy

Ax

Ay

In 3D



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Ax

Ay =

R xx Rxy

R yx R yy

Ax

Ay

In 3D

A

xAy

Az

=R xx R xy R xz

R yx R yy R yz

R zx R zy R zz

Ax

Ay

Az



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Ax

Ay =

R xx Rxy

R yx R yy

Ax

Ay

In 3D

A

xAy

Az

=R xx R xy R xz

R yx R yy R yz

R zx R zy R zz

Ax

Ay

Az

Transformation of Vector under Rotation of Axes

[A ] = [R ] × [A]



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Ax

Ay =

R xx Rxy

R yx R yy

Ax

Ay

In 3D

Ax

Ay

Az

=R xx R xy R xz

R yx R yy R yz

R zx R zy R zz

Ax

Ay

Az

Transformation of Vector under Rotation of Axes

[A ] = [R ] × [A]

Denition: A Physical Quantity which transforms under rotation as

above is a VectorVectors Advanced Topics 9/9

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Lec1 Vectors

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