Transcript Ch33

Chapter 3: Vectors
Chapter 3 Goals:
• To introduce the concepts and notation for vectors:
quantities that, in a single package, convey both
magnitude and direction
• To be able to visualize vectors, and perform arithmetic
operations upon them (addition and subtraction)
• To define and make use of unit vectors
• To become able to use this language of magnitude and
direction, as contrasted with the usage of component
language, especially in two dimensions
• To understand the paradigmatic vector: the
displacement vector
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Obvious language for vectors in two dimensions:
magnitude and direction
A
\A\
y
qA
• magnitude of A: written
|A| (some books say A)
• it is just the length of A
• |A| is always positive or 0
• technically it is a scalar
because it doesn’t change if
you rotate your coordinates
• |A| carries the units of A
x
• direction of A: written qA
• usually counterclockwise from x axis
• units of qA are degrees or radians
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Polar coordinates in 2 dimensions
• (x,y) are cartesian coordinates
• two numbers needed to
specify full information
• convention for q is to start
along x and swing ccw
r  x 2  y 2  distance (only)
r   hypotenuse
x   adjacent
y   opposite
• r : a scalar because coordinate system’s orientation does
not affect r
• x and y : not, technically, scalars, but they are numbers
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Polar coordinates in 2 dimensions
r   hypotenuse
x   adjacent
y   opposite
• Upshot of this: 2d vector
arithmetic looks very much
like working on positions in
polar coordinates
cos q  a  x  r cos q
h
sinq  o  y  r sinq
h
tan q  o 
a
q  Arctan  y x 
 180 if x  0]
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r  x2  y2
Pros of magnitude-direction language
• we can write A = {|A|,q} = {A, q}
• lends itself to obvious pictures
• can easily be converted to compass (map) language:
East   x, and North   y
• addition and subtraction of vectors is done pictorially
Cons of magnitude-direction language
• accuracy of pictorial addition and subtraction is
limited to human ability with protractor and ruler
• laws of sines and cosines needed to calculate: messy
• only convenient in 2d!! 3d requires the dreaded
spherical trigonometry because there are 2 angles!!
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
The displacement vector: we move ‘where’ into
a higher dimension at last!
• initial position is ri
y
• final position is rf
Dr
• displacement is
Dr:= rf – ri
rf
• we are subtracting!! Ouch!!
ri • notice : the positions are
origin-dependent, but the
x
displacement is not!!
• the two position vectors are drawn in standard position,
which means their tail is at the origin: makes sense
• the displacement vector is not drawn in s.p.
• if it behaves like the displacement vector, then it is a
vector!! It is the ‘paradigmatic’ vector.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Triangle method for adding vectors
• A (and B) are two
displacement vectors
with B following A
• magnitude : |A| or A
qB
qR
• direction: an angle qA,
which looks to be 0°
• we add to it vector B, in triangle method: tip-to-tail
• B’s angle qB looks to be about 60°
• A is added to B, to give resultant R = A + B
• lay down A and B with ruler and protractor; draw R
• R’s angle qR looks to be about 40°
• with A in s.p., B is not, but R is in s.p.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Some more facts about vectors
• A + B = B + A: commutative
• (A + B) + C = A + (B + C): associative
• the negative of a vector is – A: same length as A but
opposite direction [additive inverse]
• There is a zero vector (no length)
• vectors carry units; when added, units must be the same
• cA: also a vector (in opposite direction if c < 0) but
‘rescaled’ in length: |cA| = |c| |A|
• to add more than two
vectors, just generalize
the triangle method, tip
to tail style
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How to subtract one vector from another
{show Active
Figure AF_0306}
• A – B := A + (– B): to subtract, just
add the additive inverse
• or: put both A and B tail-to-tail:
then C = A – B has its tail at the tip of
B (the one subtracted) and has its tip at
the tip of A (the one added)
• in other words, we can see that we
have added C to B in the additive
‘triangle’ way, to arrive at A
• Cseems to ‘start’ at B and end at A
• Example: the displacement vector is
precisely the change in the position
vector!!
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Addition Example : Taking a Hike
A hiker begins a trip by
first walking 25.0 km
southeast from her car.
She stops and sets up her
tent for the night. On the
second day, she walks
40.0 km in a direction
60.0° north of east, at
which point she discovers
a forest ranger’s tower.
• Using ruler and protractor, lay out the two displacements
A and B. Then, with the same tools, measure R.
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A different and more flexible language:
component language (like polars)
Ax y
A
|A|
• draw vector with tail at origin:
standard position
•“drop a perpendicular” from
Ay
tip to either Cartesian axis
qA • you have made a right triangle
•in this sketch, qA is larger than 90
but you can deal…
x • |A| is the hypotenuse; |A| ≥ 0
• Ax is the adjacent and here < 0
• Ay is the opposite and here > 0
• we write A = < Ax , Ay >
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Converting from magnitude-angle to components
cos q A  a 
h
Ax
A
 Ax  A cos q A
Ay
o
sinq A 

 Ay  A sinq A
h
A
Converting from magnitude-angle to components
A
Ax2

Ay2
tanq A 
Ay
Ax
{show Active Figure AF_0303}
Aspects of component language
• addition (subtraction ) of vectors is simplicity itself: add
(subtract) the components as numbers!
• components are numbers but technically not scalars
since components are different in a rotated coordinate
system
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Revisiting the vector addition example
 
 25 km sin315   25 km - .707  17.7 km
Ax  A cos q A  25 km cos 315  25 km .707  17.7 km
Ay  A sinq A

 
 40 km sin60   40 km .866  34.6 km
Bx  B cos q A  40 km cos 60  40 km .500  20.0 km
B y  B sinq A

 Rx  Ax  Bx  17.7 km  20 km
 37.7 km
 R y  Ay  B y  17.7 km  34.6 km
 16.9 km
R  37.7 km,16.9 km 
Section 3.4
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Finishing the Example
• Now we would work back from component language
to magnitude-angle language:
R  Rx2  R y2 

37.7 km2  16.9 km2
 1420 km  286 km
tan q R 
2
Ry
Rx
2

1
2
 16.9 km

 1706 km
2

1
2
37.7 km
 24.1
 .448  q R  Arctan .448  
204.1
• ‘net displacement is 41.3 km, at a
bearing of 65.9° East of North’
• R = {41.3 km, 24.1°}
Section 3.4
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 41.3 km
A third language that uses components:
unit vectors
• i is a vector of unit length, with
i Ax
A
|A|
no units, that points along x
• j and k are similar, along y (and z)
j Ay • create the vectors i Ax & j Ay
y
qA
A  ˆi Ax  ˆj Ay (kˆ Az )
x
• A truly explicit way to write A
• remember: |i| = |j| = |k| = 1
• one unit long, but carry no units,
so the name ‘unit vector’ is dumb
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What other operations can we do with vectors?
• cannot divide by a vector; vectors are only ‘upstairs’
• dot (scalar (inner)) product of two
• cross (vector (outer (wedge))) product of two
• we can take their derivative with respect to a scalar
• we can integrate them but usually we integrate some
kind of scalar product…
The scalar product of two vectors
A
• put them tail-to-tail, with q the
angle between (0° ≤ q ≤ 180°)
 A  B :
q
B
A B cosq
• the result is indeed a scalar
• A∙B = B ∙A (commutative)
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More about the scalar product
• a measure of the parallelness of the vectors, as well
as the magnitudes
• A∙B = (|A|)(|B| cos q) = length of A times the length
of B’s projection along the line of A
• A∙B = (|B|)(|A| cos q) = length of B times the length
of A’s projection along the line of B
• A∙A = |A|2
• i∙i = j∙j = k∙k =1 i∙j = j∙k = 0 etc.
• a vector’s component in a certain direction is the
scalar product of that vector with a unit vector in that
direction: Cn = C∙n
• [to make a unit vector, just divide a vector by its
magnitude: a = A/|A| ]
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