1 Vectors and Polar Coordinates Lecture 2 (04 Nov 2006) Enrichment Programme for Physics Talent 2006 /07 Module I
Dec 19, 2015
1
Vectors and Polar Coordinates
Lecture 2 (04 Nov 2006)
Enrichment Programme for Physics Talent 2006/07Module I
2
2.1 Vectors and scalars2.2 Matrix operations of
rotations2.3 Polar coordinates
3
Vector: quantity having both magnitude and direction, e.g., displacement, velocity, force, acceleration, …
Scalar: quantity having magnitude only, e.g., mass, length, time, temperature, …
2.1 Vectors and scalars
A
4
Fundamental definitions:
Two vectors and are equal if they have the same magnitude and direction regardless of the initial points
Having direction opposite to but having the same magnitude
2.1 Vectors and scalars
A
B
A
B
A
A
A
5
Addition:
subtraction:
2.1 Vectors and scalars
C A B
A
B
A B
A
C
A
B
C A B
B
C
6
Laws of vector
2.1 Vectors and scalars
1.
2.
3.
4.
( ) ( )
( ) ( )
(5.
)
(6. )
A B B A
A B C A B C
A A
A A
A A A
A B A B
7
Null vector: vector with magnitude zero
Unit vector: vector with unit magnitude, i.e., .
Rectangular unit vectors , and .
, (x, y, z) are different components of the vector .Magnitude of : 2 2 2A x y z
2.1 Vectors and scalars
aA A
unit vector
i j k
i
j
kˆˆ ˆA xi yj zk
A
A
8
Example: Find the magnitude and the unit vector of a vector ˆˆ ˆ2A i j k
Magnitude: 2 2 2( 1) 2 ( 1) 6A
Unit vector:
1 2 1 ˆˆ ˆˆ6 6 6
Aa i j k
A
ˆA Aa
Write: , where
2.1 Vectors and scalars
9
Dot and cross productDot product: , where is the angle between vectors and .
Laws of dot product:
2.1 Vectors and scalars
cosA B AB
A
B
1 2 3 1 2 3
1 1 2 2 3 3
( )
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ1 0
ˆ ˆˆ ˆ ˆ ˆ
;
and
1.
2.
3.
4.
A B B A
A B C A B A C
i i j j k k i j j k k i
A Ai A j A k B B i B j B k
A B A B A B A B
A
B
10
2. and
Example: Evaluate the dot product of vectors ˆˆ ˆ2A i j k
1. and ˆˆ ˆ2 3B i j k
ˆˆ ˆ 3A i j k ˆˆ ˆ3 2B i j k
2 6 1 5A B
1.
2. 1 3 6 4A B
2.1 Vectors and scalars
11
Dot and cross product
cross product: , where is the angle between vectors and . is a unit vector such that , and form a right-handed system.
2.1 Vectors and scalars
ˆ sinA B cAB
A
B
cA
B
c
A
B
ˆ sincAB
A
B
area of the parallelogram
12
Dot and cross product
Laws of cross product:
2.1 Vectors and scalars
( )
ˆ ˆˆ ˆ ˆ ˆ 0
ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ
A B B A
A B C A B A C
i i j j k k
i j k j k i k i j
;
, ,
1.
2.
3.
i
j
k
13
2. and
Example: Evaluate the cross product of vectors
ˆˆ ˆ2A i j k
1. and ˆˆ ˆ2 3B i j k
ˆˆ ˆ 3A i j k ˆˆ ˆ3 2B i j k
ˆˆ ˆ3 7A B i j k
1.
2. ˆˆ ˆ11 4A B i j k
2.1 Vectors and scalars
14
2.2 Matrix operations of rotations
a vector in a 2-dimensional plane can be written as ,
and are called the basis vector, since any vector can be written as a linear combination of the basis vector
1 2ˆ ˆ
v v i v j
i j
i
j(v1, v2)
Vectors in 2-dimensions
15
2.2 Matrix operations of rotations
any vector in R2 can be written as
and are called the base vectors, since any vector can be written as a linear combination of the base vectors, namely
Is base vectors unique?
1 2ˆ ˆ
v v i v j
i j
Vectors in 2-dimensions
1 2ˆ ˆ
v v i v j
base vectors are not unique!
i
ˆ 'j
(v1’, v2
’)ˆ 'i
(v1, v2)j
16
Hence, and are example of orthonormal base vectors.
Generally, let and are base vectors, i.e. 1 1 2 2ˆ ˆ v v e v e
1e 2e
Base vectors are said to be orthonormal if
1 1 2 2
1 2
ˆ ˆ ˆ ˆ 1
ˆ ˆ 0
e e e e
e e
i j
2.2 Matrix operations of rotationsVectors in 2-dimensions
17
1e
2ˆe
(v1’, v2
’)1e
(v1, v2)
2eLet both and are orthonormal base vectors, i.e.,
1 2ˆ ˆ( , )e e 1 2ˆ ˆ( , ) e e
11 1 2 2 1 22ˆ ˆ ˆˆ v v e e vev ev
using different coordinate system to represent is possible.since 1 1 1 2 2
2 1 1
1 1 1
2 222
ˆ ˆ
ˆ ˆ
ˆ ˆ
ˆ ˆ
v v e e
e e
e v e
v v e v eHow to express them in matrix
form?
2.2 Matrix operations of rotationsVectors in 2-dimensions
18
1 1 2 1
2 1 2 2
1 1 1
2
1 1
2 2
1 1
2 2
2
ˆ ˆ
ˆ ˆ
cos sin
sin cos
cos sin
sin cos
ˆ ˆ
ˆ ˆ
v e e v
v e e v
v v
v v
v v
v v
e e
e e
or in matrix form:
Note are orthogonal.
2.2 Matrix operations of rotationsVectors in 2-dimensions
2ˆe
(v1’, v2
’)1e
(v1, v2)
2e
1e
cos sin cos sin
sin cos sin cos
and
19
1 1
2 2
v vR
v v
Hence, an orthogonal matrix R acts as transformation to transforms a vector from one coordinates to another, i.e.,
2.2 Matrix operations of rotationsVectors in 2-dimensions
20
2.3 Polar coordinates
The position of the “Red Point” can be represented by (r, ) instead of (x, y) in Cartesian Coordinates.
r = magnitude of the position vector r
= angle of the position vector and the x-axis x
y
O
21
2.3 Polar coordinates
r
In Polar Coordinates, we define two new base vectors instead of in Cartesian Coordinates.
ˆˆ,r ˆ ˆ,i j
: a unit vector in the direction of increasing r (i.e. -direction)
r
r
: a unit vector in the direction of increasing
y
xO
22
2.3 Polar coordinates
r
Any vector on the 2D plane can be expressed in terms of and :
y
x
r
ˆˆrV V r V
In particular, the position vector is given by
r
ˆr rr
O
V
rV
V
23
2.3 Polar coordinates
Conversions between Polar Coordinates (r, ) and Cartesian Coordinates (x, y):
Cartesian Coordinates:
,x y , ,x y
Cylindrical Coordinates:
,r 0, 0 2r
24
2.3 Polar coordinates
Conversions between Polar Coordinates (r, ) and Cartesian Coordinates (x, y):
:, ,r x y cos ,x r siny r
:
2 2r x y tany
x
:, ,r x y
25
2.3 Polar coordinates
Conversions between Polar Coordinates (r, ) and Cartesian Coordinates (x, y):
:
ˆ ˆcos sinˆ ir j
ˆ ˆsin cosˆ i j
26
2.3 Polar coordinates
ˆˆrV V r V
Differentiating a vector in Polar Coordinates (r, ):
:
ˆˆ ˆˆrr
dVdVdV dr dr V V
dt dt dt dt dt
ˆ ˆˆ ˆsin cosdr d d d
i jdt dt dt dt
ˆˆ rr
dVdVdV d dr V V
dt dt dt dt dt
ˆˆ ˆ ˆcos sin
d d d di j r
dt dt dt dt
27
2.3 Polar coordinates
Central Force Field Problem:
ˆ( ) ( )F r F r r
ˆ( ) ( ) 0dL
N r F r r rF rdt
External Torque = 0:
Conservation of Angular Momentum L
28
Recall: momentum , where m is the mass, is a measure of the linear motion of an object.
The angular momentum of an object is defined as: a measure of the rotational motion of an object.
Box 2.1 Angular momentum
L r p
p mv
r
p
29
As linear momentum, an object keeps its motion unless an external force is acted;
An object has a tendency to keep rotating unless external torque is acted. It is the conservation of angular momentum.
Box 2.1 Angular momentum
The conservation of angular momentum explains why the Earth always rotates once every 24 hours.
30
Area swept out in a very small time interval:
m
L
t
A
tvmrm
A
tvrA
2
2
12
1
2.3 Polar coordinates
31
2.3 Polar coordinates In general, planets’ orbits are elliptical To describe its motion,
r
v
32
is constant if angular momentum
is conserved and m is unchanged.m
L
t
A
2
2.3 Polar coordinates
33
34
This is in fact one of his famous three laws of planetary motion, which are deduced from Tycho’s 20 years observation data.
Johannes Kepler ( 開普勒 ) 1571 - 1630
2.3 Polar coordinates
35
The second law of planetary motion: equal time sweeps equal area
closer to the sun, planet moves faster
farther away from the sun, planet moves slower
2.3 Polar coordinates
36
Coordinates Systems in 3D Space
Cartesian Coordinates:
, , ,x y z
37
Coordinates Systems in 3D Space
Cylindrical Coordinates:
0, 0 2 , ,z
38
Coordinates Systems in 3D Space
Spherical Coordinates:
0, 0 2 , 0r