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Transcript of 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
1 Physical Quantities
2 Operations with Vectors
3 Cartesian Coordinates
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Lecture 1
VECTORS
1 Physical Quantities
2 Operations with 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
Vectors Physical Quantities 2/9
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Physical Quantities
Scalars : described by a single number (magnitude)
Mass, Temperature, SpeedVectors : direction & magnitude
Velocity, Acceleration, Angular Momentum, Electric, Magnetic Field
Vectors Physical Quantities 2/9
8/10/2019 Lec1 Vectors
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Physical Quantities
Scalars : described by a single number (magnitude)
Mass, Temperature, Speed
Vectors : direction & magnitude
Velocity, Acceleration, Angular Momentum, Electric, Magnetic Field
Vectors are
Vectors Physical Quantities 2/9
8/10/2019 Lec1 Vectors
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Physical Quantities
Scalars : described by a single number (magnitude)
Mass, Temperature, Speed
Vectors : direction & magnitude
Velocity, Acceleration, Angular Momentum, Electric, Magnetic Field
Vectors are
Arrows in Space
Vectors Physical Quantities 2/9
8/10/2019 Lec1 Vectors
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Physical Quantities
Scalars : described by a single number (magnitude)
Mass, Temperature, Speed
Vectors : direction & magnitude
Velocity, Acceleration, Angular Momentum, Electric, Magnetic Field
Vectors are
Arrows in Spacedenoted (always!) by a symbol with anarrow sign on top:
−→A
Vectors Physical Quantities 2/9
8/10/2019 Lec1 Vectors
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Physical Quantities
Scalars : described by a single number (magnitude)
Mass, Temperature, Speed
Vectors : direction & magnitude
Velocity, Acceleration, Angular Momentum, Electric, Magnetic Field
Vectors are
Arrows in Spacedenoted (always!) by a symbol with anarrow sign on top:
−→A
Magnitude of−→A denoted as |
−→A | or
simply A
Vectors Physical Quantities 2/9
8/10/2019 Lec1 Vectors
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Physical Quantities
Scalars : described by a single number (magnitude)
Mass, Temperature, Speed
Vectors : direction & magnitude
Velocity, Acceleration, Angular Momentum, Electric, Magnetic Field
Vectors are
Arrows in Spacedenoted (always!) by a symbol with anarrow sign on top:
−→A
Magnitude of−→A denoted as |
−→A | or
simply A
Tensors : Generalization of vectors:
Vectors Physical Quantities 2/9
8/10/2019 Lec1 Vectors
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Physical Quantities
Scalars : described by a single number (magnitude)
Mass, Temperature, Speed
Vectors : direction & magnitude
Velocity, Acceleration, Angular Momentum, Electric, Magnetic Field
Vectors are
Arrows in Spacedenoted (always!) by a symbol with anarrow sign on top:
−→A
Magnitude of−→A denoted as |
−→A | or
simply A
Tensors : Generalization of vectors: products of vectors
Vectors Physical Quantities 2/9
8/10/2019 Lec1 Vectors
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Physical Quantities
Scalars : described by a single number (magnitude)
Mass, Temperature, Speed
Vectors : direction & magnitude
Velocity, Acceleration, Angular Momentum, Electric, Magnetic Field
Vectors are
Arrows in Spacedenoted (always!) by a symbol with anarrow sign on top:
−→A
Magnitude of−→A denoted as |
−→A | or
simply A
Tensors : Generalization of vectors: products of vectors
Moment of Inertia, Stress tensor, permeability, energy-momentum
Vectors Physical Quantities 2/9
h
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Operations with Vectors
Vectors Operations with Vectors 3/9
O i i h V
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Operations with Vectors
Addition:−→
A +−→
B =−→
C
Vectors Operations with Vectors 3/9
O i i h V
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Operations with Vectors
Addition:−→
A +−→
B =−→
C
Multiplication by scalar:
a ×−→A =
−→aA
Vectors Operations with Vectors 3/9
O ti ith V t
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Operations with Vectors
Addition:−→
A +−→
B =−→
C
Multiplication by scalar:
a ×−→A =
−→aA
Scalar Product:−→A ·
−→B = AB cos θ
Vectors Operations with Vectors 3/9
O ti ith V t
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Operations with Vectors
Addition:−→
A +−→
B =−→
C
Multiplication by scalar:
a ×−→A =
−→aA
Scalar Product:−→A ·
−→B = AB cos θ
Cross Product −→A ×
−→B =
−→C
Vectors Operations with Vectors 3/9
Vectors: Component form
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Vectors: Component form
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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Vectors: Component form
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Vectors: Component form
Description relative to a set of coordinates in 3d Space.
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
Vectors Cartesian Coordinates 4/9
Vectors: Component form
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Vectors: Component form
Description relative to a set of coordinates in 3d Space.
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
ˆi ·
ˆi =
ˆ j ·
ˆ j =
ˆk ·
ˆk = 1
Vectors Cartesian Coordinates 4/9
Vectors: Component form
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Vectors: Component form
Description relative to a set of coordinates in 3d Space.
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
ˆi ·
ˆi =
ˆ j ·
ˆ j =
ˆk ·
ˆk = 1
i · j = j · k = k · i = 0
Vectors Cartesian Coordinates 4/9
Vectors: Component form
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Vectors: Component form
Description relative to a set of coordinates in 3d Space.
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
ˆi ·
ˆi =
ˆ j ·
ˆ j =
ˆk ·
ˆk = 1
i · j = j · k = k · i = 0
i × j = k ; j × k = i ;k × i = j .
Vectors Cartesian Coordinates 4/9
Vectors: Component form
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Vectors: Component form
Description relative to a set of coordinates in 3d Space.
Cartesian coordinates
$A_y$
$A_x$
$A_z$
y
z
−→A
ˆ
k
x
ˆ
i
ˆ
j
Decomposition of Vectorinto components:
Vectors Cartesian Coordinates 4/9
Vectors: Component form
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Vectors: Component form
Description relative to a set of coordinates in 3d Space.
Cartesian coordinates
y
z
−→A
Ay
Az
ˆ
k
x
Ax
ˆ
i
ˆ
j
Decomposition of Vectorinto components:Projections along theaxes
Vectors Cartesian Coordinates 4/9
Vectors: Component form
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Vectors: Component form
Description relative to a set of coordinates in 3d Space.
Cartesian coordinates
y
z
−→A
Ay
Az
ˆ
k
x
Ax
ˆ
i
ˆ
j
Decomposition of Vectorinto components:Projections along theaxes
−→A = Ax i + A y j + A z k
Vectors Cartesian Coordinates 4/9
Vectors: Component form
8/10/2019 Lec1 Vectors
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p
Description relative to a set of coordinates in 3d Space.
Cartesian coordinates
y
z
−→A
Ay
Az
ˆ
k
x
Ax
ˆ
i
ˆ
j
Decomposition of Vectorinto components:Projections along theaxes
−→A = Ax i + A y j + A z k
i , j , k are constantvectors.
Vectors Cartesian Coordinates 4/9
Operations with Vectors: Component form
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p p
Vectors Cartesian Coordinates 5/9
Operations with Vectors: Component form
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p p
Magnitude of−→A : A = A2
x + A 2y + A 2
z
Vectors Cartesian Coordinates 5/9
Operations with Vectors: Component form
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p p
Magnitude of−→A : A = A2
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
Vectors Cartesian Coordinates 5/9
Operations with Vectors: Component form
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Magnitude of−→A : A = A2
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
Vectors Cartesian Coordinates 5/9
Operations with Vectors: Component form
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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
Vectors Cartesian Coordinates 6/9
Operations with Vectors: Component form
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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
Direction of vector product: right hand screw rule
Vectors Cartesian Coordinates 6/9
Operations with Vectors: Component form
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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
Direction of vector product: right hand screw rule
−→A ×
−→B
−→A
−→B
θ
Vectors Cartesian Coordinates 6/9
Time Rate of Change of a Vector
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O
Vectors Time rate of change of a vector 7/9
Time Rate of Change of a Vector
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O
Vectors Time rate of change of a vector 7/9
Time Rate of Change of a Vector
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O
trajectory
Vectors Time rate of change of a vector 7/9
Time Rate of Change of a Vector
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O
r(t)
trajectory
Vectors Time rate of change of a vector 7/9
Time Rate of Change of a Vector
8/10/2019 Lec1 Vectors
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O
r(t)r (t + ∆ t)
trajectory
Vectors Time rate of change of a vector 7/9
Time Rate of Change of a Vector
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O
r(t)r (t + ∆ t)
∆ r
trajectory
Vectors Time rate of change of a vector 7/9
Time Rate of Change of a Vector
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O
r(t)r (t + ∆ t)
∆ r
trajectory lim∆ t → 0
−→∆
r∆ t =
Vectors Time rate of change of a vector 7/9
Time Rate of Change of a Vector
8/10/2019 Lec1 Vectors
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O
r(t)
r (t + ∆ t)
∆ r
trajectory lim∆ t → 0
−→∆
r∆ t = d−→
rdt =
Vectors Time rate of change of a vector 7/9
Time Rate of Change of a Vector
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O
r(t)
r (t + ∆ t)
∆ r
trajectory lim∆ t → 0
−→∆
r∆ t = d−→
rdt = dxdt i + dydt j + dzdt k
Vectors Time rate of change of a vector 7/9
Time Rate of Change of a Vector
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O
r(t)
r (t + ∆ t)
∆ r
trajectory lim∆ t → 0
−→∆
r∆ t = d−→
rdt = dxdt i + dydt j + dzdt k
d−→A
dt =
Vectors Time rate of change of a vector 7/9
Time Rate of Change of a Vector
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O
r(t)
r (t + ∆ t)
∆ r
trajectory lim∆ t → 0
−→∆
r∆ t = d−→
rdt = dxdt i + dydt j + dzdt k
d−→A
dt =
dAx
dti +
dAy
dt j +
dA z
dtk
Vectors Time rate of change of a vector 7/9
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
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
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x
y
A
Vectors Advanced Topics 8/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
8/10/2019 Lec1 Vectors
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x
y
AAy
Ax
Vectors Advanced Topics 8/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
8/10/2019 Lec1 Vectors
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y ′
x ′
x
y
AAy
Ax
θ
Vectors Advanced Topics 8/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
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y ′
x ′
x
y
AAy
Ax
A ′
y A ′
xθ
Vectors Advanced Topics 8/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
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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
Vectors Advanced Topics 8/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
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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
Vectors Advanced Topics 8/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
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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
AxAy
Vectors Advanced Topics 8/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
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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
AxAy
= cosθ sin θ− sin θ cosθ
Vectors Advanced Topics 8/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
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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
AxAy
= cosθ sin θ− sin θ cosθ AxAy
Vectors Advanced Topics 8/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
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Ax
Ay
Vectors Advanced Topics 9/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
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Ax
Ay =
R xx Rxy
R yx R yy
Vectors Advanced Topics 9/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
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Ax
Ay =
R xx Rxy
R yx R yy
Ax
Ay
Vectors Advanced Topics 9/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
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Ax
Ay =
R xx Rxy
R yx R yy
Ax
Ay
In 3D
Vectors Advanced Topics 9/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
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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
Vectors Advanced Topics 9/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
8/10/2019 Lec1 Vectors
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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]
Vectors Advanced Topics 9/9
Advanced Topic 1Transformation of Vectors under Rotation of Coordinate Axes
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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