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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