PHY 113 A Fall 2012 -- Lecture 4 19/5/2012

PHY 113 A General Physics I9-9:50 AM MWF Olin 101

Plan for Lecture 4:

Chapter 3 – Vectors

1. Abstract notion of vectors

2. Displacement vectors

3. Other examples

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PHY 113 A Fall 2012 -- Lecture 4 39/5/2012

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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?

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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 A Fall 2012 -- Lecture 4 69/5/2012

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 A Fall 2012 -- Lecture 4 79/5/2012

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.

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Vector addition:

ab

a – b

Vector subtraction:

a

-b

a + b

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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

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Vector components:

ax

ay

22yx aa a

jiyxa ˆˆˆˆ yxyx aaaa

yxba

yxbyxa

ˆˆ

ˆˆ and ˆˆFor

yyxx

yxyx

baba

bbaa

PHY 113 A Fall 2012 -- Lecture 4 119/5/2012

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

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Vector components; using trigonometry

An orthogonal coordinate system

A

kjizyxA ˆˆˆˆˆˆ zyxzyx AAAAAA

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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 A Fall 2012 -- Lecture 4 149/5/2012

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 A Fall 2012 -- Lecture 4 159/5/2012

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

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Example of vector addition:

a

ba + b

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a

ba + b

gcos2

:Dallas and Chicagobetween Distance

22bababa

oooo 7421590 g

mi 788bag

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Webassign version:

f

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

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Another example:

ji

BAR

jiB

jiA

ˆ9.16ˆ7.37

ˆ6.34ˆ0.20

ˆ7.17ˆ7.17

:units kmin ectorsPosition v