9/3/2013PHY 113 C Fall 2013 -- Lecture 31 PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors 1.Abstract notion.

PHY 113 C Fall 2013 -- Lecture 3 19/3/2013

PHY 113 C General Physics I11 AM-12:15 PM TR Olin 101

Plan for Lecture 3:

Chapter 3 – Vectors

1. Abstract notion of vectors

2. Displacement vectors

3. Other examples

PHY 113 C Fall 2013 -- Lecture 3 29/3/2013

PHY 113 C Fall 2013 -- Lecture 3 39/3/2013

PHY 113 C Fall 2013 -- Lecture 3 49/3/2013

iclicker question

A. Have you attended a tutoring session yet?B. Have you attended a lab session yet?C. Have you attended both tutoring and lab sessions?

PHY 113 C Fall 2013 -- Lecture 3 59/3/2013

Mathematics Review -- Appendix B Serwey & Jewett

iclicker question

A. Have you used this appendix?B. Have you used the appendix, and find it helpful?C. Have you used the appendix, but find it unhelpful?

iclicker question

Have you changed your webassign password yet?A. yesB. no

PHY 113 C Fall 2013 -- Lecture 3 69/3/2013

Question from Webassign #2

1 2 3 4 5 6 7 8 9 10

8

46

2

-8-6 -4-20

PHY 113 C Fall 2013 -- Lecture 3 79/3/2013

Mathematics Review -- Appendix B Serwey & Jewett

a

acbbx

cbxax

2

4

0

:equation Quadratic

2

2

)cos()sin(

:calculus alDifferenti

1

ttdt

d

eedt

d

antatdt

d

tt

nn

)cos(1

)sin(

11

:calculus Integral1

tdtt

edte

n

atdtat

tt

nn

a

b

c

q

b

ac

ac

b

tan

sin

cos

:ryTrigonomet

PHY 113 C Fall 2013 -- Lecture 3 89/3/2013

Definition of a vector

1. A vector is defined by its length and direction.

2. Addition, subtraction, and two forms of multiplication can be defined

3. In practice, we can use trigonometry or component analysis for quantitative work involving vectors.

4. Abstract vectors are useful in physics and mathematics.

PHY 113 C Fall 2013 -- Lecture 3 99/3/2013

Vector addition:

ab

a – b

Vector subtraction:

a

-b

a + b

PHY 113 C Fall 2013 -- Lecture 3 109/3/2013

Some useful trigonometric relations

(see Appendix B of your text)

g

c

b

a

a

b Law of cosines:

a2 = b2 + c2 - 2bc cos

b2 = c2 + a2 - 2ca cos

c2 = a2 + b2 - 2ab cos g

Law of sines:

sin c

sin b

sin a

g

PHY 113 C Fall 2013 -- Lecture 3 119/3/2013

Some useful trigonometric relations -- continued

(from Appendix B of your text)

g

c

b

a

a

b Law of cosines:

a2 = b2 + c2 - 2bc cos

b2 = c2 + a2 - 2ca cos

c2 = a2 + b2 - 2ab cos g

PHY 113 C Fall 2013 -- Lecture 3 129/3/2013

Some useful trigonometric relations -- continued

Example:

20o

c=?

15

a

10

b Law of cosines:

a2 = b2 + c2 - 2bc cos

b2 = c2 + a2 - 2ca cos

c2 = a2 + b2 - 2ab cos g

654.60922.43

0922.4320cos151021510 222

c

c o

PHY 113 C Fall 2013 -- Lecture 3 139/3/2013

Possible realization of previous example:

20o

c=?

15m

a

10m

b

Start

South

East

A pirate map gives directions to buried treasure following the indicated arrows. A wily physics students decides to take the easterly direct route after computing the distance c.

treasure

PHY 113 C Fall 2013 -- Lecture 3 149/3/2013

Quantitative representation of a vector

Cartesian coordinates

i

j

A

Ax

Ay

jiA ˆˆyx AA

PHY 113 C Fall 2013 -- Lecture 3 159/3/2013

Quantitative representation of a vector

reference direction

AA

q

Note: q can be specified in degrees or radians; make sure that your calculator knows your intentions!

Polar coordinates

PHY 113 C Fall 2013 -- Lecture 3 169/3/2013

Quantitative representation of a vector

AA

q

Polar & cartesian coordinates

icosAAx

sinAAy

x

y

A

A

A

A

tancos

sin:note Also

PHY 113 C Fall 2013 -- Lecture 3 179/3/2013

Vector components:

ax

ay

22yx aa a

jiyxa ˆˆˆˆ yxyx aaaa

yxba

yxbyxa

ˆˆ

ˆˆ and ˆˆFor

yyxx

yxyx

baba

bbaa

PHY 113 C Fall 2013 -- Lecture 3 189/3/2013

ax

ay

q

y

a = 1 m

Suppose you are given the length of the vector a as shown. How can you find the components?

A. ax=a cos , q ay=a sin qB. ax=a sin , y ay=a cos yC. Neither of theseD. Both of these

iclicker question

PHY 113 C Fall 2013 -- Lecture 3 199/3/2013

Vector components; using trigonometry

An orthogonal coordinate system

A

kjizyxA ˆˆˆˆˆˆ zyxzyx AAAAAA

PHY 113 C Fall 2013 -- Lecture 3 209/3/2013

Vector components:

ax

ay

yxba

yxbyxa

ˆˆ

ˆˆ and ˆˆFor

yyxx

yxyx

baba

bbaa

yxa ˆˆ yx aa

by

bx

yxb ˆˆ yx bb ba

PHY 113 C Fall 2013 -- Lecture 3 219/3/2013

Examples

Vectors Scalars

Position r Time t

Velocity v Mass m

Acceleration a Volume V

Force F Density m/V

Momentum p Vector components

PHY 113 C Fall 2013 -- Lecture 3 229/3/2013

Vector components

zyxR ˆˆˆ1111 zyx

zyxR ˆˆˆ2222 zyx

zyxRR ˆ(ˆ)(ˆ)( )21212121 zzyyxx

Vector multiplication

“Dot” product 1ˆˆ;cos AB xxBA AB

“Cross” product zyxBA ˆˆˆ;sin|| AB AB

PHY 113 C Fall 2013 -- Lecture 3 239/3/2013

Example of vector addition:

a

ba + b

PHY 113 C Fall 2013 -- Lecture 3 249/3/2013

a

ba + b

gcos2

:Dallas and Chicagobetween Distance

22bababa

o

oooo

74

18021590

gg

mi 788bag

PHY 113 C Fall 2013 -- Lecture 3 259/3/2013

Webassign version:

f

Note: In this case the angle f is actually measured as north of east.

PHY 113 C Fall 2013 -- Lecture 3 269/3/2013

Another example:

ji

BAR

jiB

jiA

ˆ9.16ˆ7.37

ˆ6.34ˆ0.20

ˆ7.17ˆ7.17

:units kmin ectorsPosition v

PHY 113 C Fall 2013 -- Lecture 3 279/3/2013

iclicker question

A. Because physics professors like to confuse studentsB. Because physics professors like to use beautiful

mathematical concepts if at all possibleC. Because all physical phenomena can be described by

vectors.D. Because there are some examples in physics that

can be described by vectors

Why are we spending 75 minutes discussing vectors

PHY 113 C Fall 2013 -- Lecture 3 289/3/2013

Example: Vector addition of velocities

Vb

Vw

Vtotal

PHY 113 C Fall 2013 -- Lecture 3 299/3/2013

Example: Displacement in two dimensions

(0,0)

(8,5)

43.9)5()8( 22 d