Vectors and Polar Coordinates

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Vectors and Polar Coordinates

Lecture 2 (04 Nov 2006)

Enrichment Programme for Physics Talent 2006/07Module I

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2.1 Vectors and scalars2.2 Matrix operations of

rotations2.3 Polar coordinates

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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

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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


A

B

A

B

A

A

A

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Addition:

subtraction:

2.1 Vectors and scalarsC A B

A

B

A B

A

C

A

B

C A B

B

C

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Laws of vector


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

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Null vector: vector with magnitude zeroUnit 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


aA A

unit vectori j k

i

j

kˆˆ Â xi yj zk

A

A

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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 kA

Â Aa

Write: , where


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Dot and cross productDot product: , where is the angle between vectors and .Laws of dot product:


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

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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


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Dot and cross productcross product: , where is the angle between vectors and . is a unit vector such that , and form a right-handed system.


ˆ sinA B cAB

A

B

cA

B

c

A

B

ˆ sincAB

A

B

area of the parallelogram

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Dot and cross productLaws of cross product:


( )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

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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


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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

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2.2 Matrix operations of rotationsany 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, namelyIs 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

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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 if1 1 2 2

1 2

ˆ ˆ ˆ ˆ 1ˆ ˆ 0

e e e ee e

i j

2.2 Matrix operations of rotationsVectors in 2-dimensions

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1e

2ê

(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 ˆˆ ˆ

ˆ ˆˆ ˆ

v v e ee e

e v ev v e v e

How to express them in matrix

form?


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1 1 2 1

2 1 2 2

1 1 1

2

1 1

2 2

1 1

2 2

2

ˆ ˆˆ ˆ

cos sinsin cos

cos sinsin cos

ˆ ˆˆ ˆ

v e e vv e e v

v vv v

v vv v

e ee e

or in matrix form:

Note are orthogonal.


2ê

(v1’, v2

’)1e

(v1, v2)

2e

1e

cos sin cos sinsin cos sin cos

and

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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.,


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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

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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

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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

rˆr rr

O

V

rV

V

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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

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:, ,r x y cos ,x r siny r

:

2 2r x y tan yx

:, ,r x y

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:

ˆ ˆcos sinˆ ir j

ˆ ˆsin cosˆ i j

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ˆˆrV V r V

Differentiating a vector in Polar Coordinates (r, ):

:

ˆˆ ˆˆrr

dVdVdV dr dr V Vdt dt dt dt dt

ˆ ˆˆ ˆsin cosdr d d di jdt dt dt dt

ˆˆ rr

dVdVdV d dr V Vdt dt dt dt dt

ˆˆ ˆ ˆcos sind d d di j r

dt dt dt dt

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2.3 Polar coordinates Central Force Field Problem:

ˆ( ) ( )F r F r r

ˆ( ) ( ) 0dLN r F r r rF rdt

External Torque = 0:

Conservation of Angular Momentum L

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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

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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.

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Area swept out in a very small time interval:

mL

tA

tvmrm

A

tvrA

2

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21


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2.3 Polar coordinates In general, planets’ orbits are elliptical To describe its motion,

r

v

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is constant if angular momentum is conserved and m is unchanged.

mL

tA

2


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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


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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


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Coordinates Systems in 3D Space

Cartesian Coordinates:

, , ,x y z

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Cylindrical Coordinates:

0, 0 2 , ,z

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Spherical Coordinates:

0, 0 2 , 0r

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Vectors and Polar Coordinates