bad news today is that there will be
quite a bit of math
but the good news is that we'll only do
it once
and it'll only take
something like although our
uh... they are quantities in physics
which i determined uniquely by one
number
massey is one of them
temperatures one of them
speed is one of them week although
skaters
there are others where you need more
than one number
went and all the one i mentioned or
motion velocity
it has a certain magnitude that's the
speech
but you also have to know what it goes
this way or that play
there has to be a direction
velocity
is a vector
acceleration
is a factor
and today
we're going to learn how to work
which these
vacuous
of bacteria
has length
and the back door
has a direction and that's why we
actually represented by
ero
you all have seen
is perfect
remember this
this is a vector
if you look at the vector head on the
issue dot
if you look at the vector
from behind if you cross
this piece
they vector and that will be our
representation
vectors
imagine that i'm standing on
okay bowling twenty six one hundred
it at the table
and i a m spending say at point all
and i moved paloma straight line
from below two point pete
tribal like so
so that's why i am
on the table and that's where you will
see me when you look from twenty six one
hundred
it just so happens that someone is also
going to move the table
in that same amount of time
from the to their
so that means at the table
we'll have moved on
and so my point p
will have moved on
exactly the same way
and so you will see me no
at for that
you'll will see me at point asking
twenty six one hundred
although i'm still standing at the same
location
on the table to table has moved
this is not a position of the table
through the whole table has shifted
none of these two motions take play
simultaneously
then what you will see from where you're
sitting
you will see me move in twenty six one
hundred
formal straight-line to s
and this holds the
secret behind
adding all
vectors we say here
that defect or or west
whatever over it
is the vector opening
there's an arrow over it
clerks
bs
it defines how we
add vectors
they're very isolated you can
at vector x
suppose i have your factory
and i have here effectively
you can do it this way which i call that
head tail technique
i take d
and i bring it to the head of a day
so this is being
this is a factor
and then
the net result
is a pretty
is back to a c
equals
a plus
that's one way of doing it doesn't
matter whether you take me
that they will be to have a very or
whether you take
potato curry and bring it to the head of
the at the same result
there's another way you can do it
and i call it the parallel around
methods
you have a
you're doing
to tails together
it was being out
for the tailpipe torching
and now you complete this town amram
and now this
vector work
c is the same
some factory
if you have here
whichever way you prefer
you see immediately that eight clas b_
is the same as before state
there's no difference
one of the meeting all day
negative victory
well a minus eighty
equal zero
pecorelli subtracting correctly
will go
so here is back to haiti
who which vector or do i have to ad
to get zero
i have to add mine is a
welcome to use to have till technique
division eight
you have to add this vector
to have zero
claudette is mine is eight
and so my guess is nothing
but the same is a bit flipped over a
hundred eighty degrees we would use that
very often
and that brings us to the point off
subtraction
of vectors how do we
subtract sectors
so any
minus b
people seek
you know we have factory
and here we have
write this down here
and here we have back to be
one way to look at this is the following
you can safely minus b
is eight
plus
minus b
and we ought to have actors
and we know what mine is bs might you be
at the same vector
but flipped over
would you mind if the
and so the spectrum now
because any
mine is me
expectancy
eight nine
and of course you can do it in different
ways you can also think of it
ads
plus
as he previously
right you can say you can bring to the
other side you can say see frisbee
is any
seat lesbians eighteen in other words
which backdoor do i have to
ads to be to get a and they have to pay
remember grant technique again so many
ways you can do it
hatfield technique is perhaps the
easiest
and the safest
so you can add
countless number of factors
but the other annexed mom and you
finally
have the sum of five or six or seven
factors which then
can be represented by only one
when you ask a nurse for instance five
and four
then there's only one answer that is not
fight for sport is not
suppose you have to back that you have
no information on their direction
would you do know that the magnitude of
one is four and the magnitude of the
other is five at all you know
then the magnitude of the some back to
her could be nine if they are both in
the same direction that's the maximum
which could be
while if they're in opposite direction
so that you have a whole range of
possibilities
because you do not know
the direction
soviet ng and a subtraction
of activities we more complicated then
just give us
as we have seen that the
some hope
vectors can be represented by
one victory
e_-coli
it can we
take long back to earth
and we can replace it by the some of us
and recall that decomposition
of the vector
and that's going to be
very important in eight oh one
and i want you to
all of his therefore
closely
i have it back to work
which isn't three-dimensional space
this is my c-axis
it is my x-axis
y axis
and the activity of the origin of all
and he was a point heat
and i have a back door
that's effective
and what i don't know i project is back
to work
onto his three axes x_y_ and z_
there we go
it won't pass so its own method of doing
this
uh... they react
ecole is back to earth
vector a
nowadays and will be data
and it and will be fine
noted that the protection over eighty on
the white axes
has here
a number which i pull in a white
his number is eight oh xd
and his number year eulogy
gymboree projection of that specter
won't forget reactions
reno introduce
what we call you expect those
u_n_ inspectors always pointing in the
direction of the positive axes
and the u_n_ back during the accident
action
this one has a leg swamp
and rewrite for it
x roof rule always means you'll
effective
and this is the u_n_ inspector
in the white arrow action
and this is the unit back there
did you get
and now i'm going to
rewrites back to any
in terms of the three components that we
have here
put it back to haiti
i'm going to write as
a top expiring tax rules
painful y
i'm joie room
coattails easy
i'm zero
and this is a black spandex
is really effectively from the origin to
this point
telerik reporting that as a back to if
you want to this makes a defector
uses that factory
it why times also revisiting of except
this one
area why times my rule for this one
and they'll be trying to is this one
unsorted three greenback tourist added
together
i gues actually identical
to defector opie
so we have decomposed won't vector
into three
directions
and we will see that there are often
busiest of great use in eight oh one
the magnitude
of the vector f
is the square root
okay excludes
what he wants cramped
with any of these clips
and so we can
take a simple
example
for instance
i take a factory
which is just an example
feeders in action
and recording
three x roof
sohail becky three
minus five wire ok
six people
soda agreements
the three units in this direction
five units in this direction in the
miners why direction
and fixing the plus the direction
that makes a perfect or
and i call that vector
what is the magnitude of their vector
which i always write down
with vertical bars by pope toolbars
alongside
that's always the magnitude
or sometimes i simply leave the ever lol
but they'll be going on the safe side
i like this idea that you know it's
really the magnitude
becomes a scale when you do that
so that would be skrev route
of three square is mine
five chris twenty five
sixth-grade thirty-six but at this
critical seventeen
and suppose i asked you
wapis st apt
is uniquely determines of course is back
to the jury to determine did
three-dimensional space
we should be able to find flight and
beta
well
the cult final fade out
he did angle here
ninety degrees projection
sort of course i don't fade out
is in oakley divided by itself
for the call time
haider
appropriately
divided by itself which in our case
basics
divided by the screen
seventy
and you can do fine
is just simply a matter
manipulated
some numbers
we're not going to a much more difficult
parts
of sectors
and that this
multiplication of texas
we're not going to need to do this
until popped over
but i decided we might as well get it
over with now
not agreeing to do expect doors
you can add and subtract
you might have to learn about
multiplication
it sort of the job is done is like going
to the dentist is a little painful
is good for you ammon is behind you
disappears
so i'm going to talk about
multiplication of actor something that
will not come back
until october
and later in the course
they had to ways and we multiply vectors
and one leasehold
the dot product
often also called
escape
products
eight dot to be
fat dot
and up to the following
as it is a skater it affects found to be
over excuse the number
litany of why
idea why that's another number
was any of the
going to be of the that's another number
digiscape
has no longer direction
that is
the dot products
perhaps method number one that's
completed yet under the internal use
that
there is another way to find
the dot product depending upon
what you're being given or the problem
is presented to you
if someone gives you
defect or any
you have the back to be
and you happen to know these angle
between them this angle fate i which has
nothing to do with that angle theta
it's the angle between the two
the doc product
is also to following
and you may make may make an attempt to
to prove that
your project
vivek he'll be home any
uses that projection
the length of the specter
is beat
prototype
and then the doc product
if the magnitude of age
times the magnitude of b
times because i know the angle state of
the two are completely identical
well you may ask me you may say gee
uh... how do i know what data is how do
i know i should take their this angle
or maybe i should take a toe this angle
what i'm going to be making would be
makes no difference
because the coastline
over this angle here is the same as the
coastline of three hundred sixty degrees
minus data
so that makes no difference
sometimes this is faster depending upon
how the problem is
presented to you
sometimes
is faster
you can immediately see by looking at
this
easier to see them looking here
the dot product can be larger than zero
that can be equal to zero and it can be
smaller than zero
can be ironed by definition always
positive they had a magnitude
but so determined by the coast i don't
think i've a cosigner state as larger
than zero twelve minutes
larger than zero
because i'm a favorite candy zero
if the angle for trade up
spy over to another which is the two
vectors are perpendicular to each other
then adult products is zero
and it is angles they died between
ninety degrees and hundred eighty
degrees
then the coastline negative
we will see that at work
no point in flight when we're going to
deal with the work
in physics
you will see that we can do positive
work
and we can do negative work
and that has to do is a doctoral
work an energy
dot products
i could do an extremely simple
example resume
december fifth i can't think off
perhaps it's almost
insulted not meant that way
we have eight
dolby
and a
is the one that you really have on the
blackboard there
right here that day
would be
is just too wide
to light roof
that's always
well what is a dolby
a doubly
there's no axe component
so that becomes zero destroyed a color
zero
they don't know why component of the
saudis minus five
fines plus two
so i get
minus ten because there was no decompose
simple as that
was minus ten
i can give you another example
example too
supposedly itself
is the unit vector in the white election
andy
the u_n_ inspector
in the direction
than eight albeit
i'll only heard loud and clear
dab zero
there is no you don't even have to think
about anything you know to be stuart
ninety degrees
if you want to waste your time
i was substituted in here
you will see that comes out to be zero
it should work because clearly
failblog means that this
there's a big has one
that's what it means
and these z that means that
beall the because what
and all the others
thermaltake six well
i wish you luck with that
and we now go to it
whale way more difficult part of milk
occasion
and that is
vectra multiplication
which is called tobacco products
or also called
most of the time i refer to it
as the cross product
across products
is recollect so eight crosley
equals c
it's across very clear cross
and i'll tell you how i remember that is
not that number one i'm going to teach
it just like with the top product two
methods
i will tell you what method number one
which is the one that always works
spun consuming
always works
you're right down here a matrix
we three rows
if ross perot
is actual
zero the second won't be effects
field y
eulogy it's important it's is here first
at that second row must be a
and the third row is pending
via fax
b_y_u_
buchanan
so these pics on numbers
v_w_ detectives
i repeated this year
verbatim
you'll see the minute why i need that
and i will do the same here
okay now comes the recipe
you go from the upper left hand corner
border one px
uh... in their direction
you multiply the mall three
and that's a plus sign
so you get a life
so see which is the cross
they crosby
equals
a life
i'm easy
by the axe roof
but i'm not going to put the actual
thing yet
because i have to sidetracked
rest well
minus time
which has
eighteen b y
sordid minus
haiti
the line
used in the direction
x
the next one
if this one
eighteen d x
whitehurst's
this one
any exp g
indirection
why
and last but not least
eighty x delight
minus
it why the act
the direction of the unit
so this part here
is what we call to see you back sixty x
components
we'll be back to it
and basically conchal field y
and this week and call
group c
you can also write that back to them
that city
people steal backs
x roof
with you why
why ru
quincy of the
cross product of a
will have lots of
exercises lots of chances you will have
on assignment two
to play which is a little bit malcolm's
my method number two
and i said number two is again as we had
with the doc product
is paid
geological
geometrical methods
let me try to
work on this board in between
if you know hector a
and you know back to be
and you know that the anguish data
then the cross product cd
equals eight crosby
is the magnitude of eighty
signed the magnitude of being
climbed his final data
because i'm afraid i as we have before
the top problem is the sign
of data
weekend already immediately see that
this will be zero if they are using a
zero degrees above eighty degrees
where's adult product was zero
wendy angle between them was ninety
degrees
his number ten the larger than zero
signed paid out of a larger than they
were can also be smaller than zero
now we only have the magnitude
of the factory
annul comes the hardest part one of the
direction of the factor
and that is something
that you have to engraved in your mind
and not forget
the direction is found as follows
you take a
because his first mentioned
and you're ok eight over the shortest
possible angle tubing
if you have in your hand of course crew
and i will show it in a minute
then you turn on the court threw a scene
from your seats clockwise
and the courts who would go into the
blackboard
and if the corkscrew goes into the
blackboard
you'll see
the tail of the back door
and you will see across
little plus sign
and airport
report that
like so
across
product is always perpendicular
to both in the
but at least you wish to torches that
can either come out the blackboard
or it can go in blackboard
and i just told you
which convention to use
and i want to show that to you
in a way that's
fuel to anymore this is what i have
used before
television helps etc sessions that i've
given to them i don't have an
appointment mom and apple visit tomato
not that that may lead to potato
i have a potato here
and here is a corkscrew
there's a corkscrew
i'm going to during the corkscrew as
seen from your site clockwise
and you'll see
that the
corkscrew ghost in to the fault take
toll
impact but i recommend of the back door
if we have to be crossing eighty
then you take do your hands annual david
over too short of angle to eight
now we have to rotate counterclockwise
andrey you rotate counterclockwise the
court threw
guns till you there you go and so
defector is not pointing in this
election and if the doctor was pointing
torch you
then we would indicate that
there's a circle
and adopt
another word for this back to her
be cross a
would have exactly the same magnitude no
difference
it would be
out of the blackboard another words
they crosby
equals minors
the crosley
there's any
double b_ is the same as b bolti
we'll encounter across products when we
deal with sports
and when we deal with
angular momentum which is not the
easiest part of a two one
let's take the next really
simple
example again
i don't mean to
insult you was such a simple example but
you will get chances more advance
chances or new assignment
i get it back to haiti
actual at the u_n_ inspector
in the excel dexion that means a package
one
it what is your role in the o'shea zero
and supposed be
is why ru
that means beale why islam
and the appraised
erected zero
and the you'll see a zero
book now
it's the
dot product across product april's me
row
you can apply that recipe
but it's much easier
go thirty x y
exceeds that we have here
they lost
eniac spelled action the u_n_ inspector
and b
in the white election
i think it is my hands
i rotate over the smallest angle which
is ninety degrees
why
and mike works will go up
so i don't know hope you're ready
i know that this cross product
must be
the magnitude was the one that's
immediately clear but i need to have the
direction by using the court's approval
now if you are very smart
you may say
hahaha
you find plus z
only because you have used it caught in
the kitchen
it this actually had been acts
and this one had been y
then the cross product annex and why
would be in the mind of the direction
yeah you're right
but if you ever do that
i will tell you
you will always always have to work with
what we call invites hamlet coordinate
system
and the right hand that coordinate
system by definition islam
whereby the cross product
of acts
with wide
is z
and not widen minor thing
so whenever you're getting the future
involved which cross products and ports
and then went in
always make yourself a x_y_z_ diagram
full rich x calls why
is the never ever make it such that
expose white is my resume
you hang yourself
well for one thing that wouldn't work
anymore
will be buried
very careful must work if you use the
right hand courts group
make sure you work
wispy
right-handed coordinate system
now the worst part is over
and now i would like to
ride on for you
we have to pick up some of the
the fruits not although it'll penetrate
slowly
i wanna ride down for you
equations for a moving particle
moving object in three-dimensional space
very complicated most u
can hardly imagine what it's like
it is a point
they're going to move around
in space and it is this point p
this point three
is going to move around in space
and i called this back to all people
i call that now vector alright
and i give it to stop index team
indicated changing with time
i call this location
why i'm going to call that
wide open
exchanging which time
according to actual fti
is going to change with time
and i called it for you
deal qty
which is going to change respond because
point he's going to move
and so i'm going to write down
the back door
is most general forum that i can do that
are which changes with time
is no actual scheme
which is the same as a opec's there
before
clients x rules
plus
wild people
viral
placentia p
zero
i've decomposed might have to all our
into three
independent back towards
each one of those
change respond
one of the velocity oldest particle
well the velocity is the first the
renovation
off the position so that is the art
bt
but illegal
forced into riveted on this one
which is the acts
bt
x roof
i'm going to write for the exp t
yeah
x dot because i'm making
and i'm going to write for a detour
activities grant x double box off and on
that note in your book
location and i will often used
but i keep wages look so clumsy
plus
why dot
time's white rule
plus developed
time zero v dot is the g_d_p_
one of the acceleration as a function of
time
well the acceleration as a function of
time
equals
d_d_t_
so does the second derivative off
extras is fine
and so that becomes
tax double dot
on tax relief
plaza wide double dot
fund why roof plus view double bill
aren't
what do we have now accomplished
it looks like my nobody's going to be
big later on
we have a point be going in
three-dimensional space
and here
the entire behavior
of the object
moves its projection along the x-axis
this is the position
this is its velocity
and diseases acceleration
and here
cd in prior behavior
on this the axis
this is the position on the z axis uses
the velocity component in the direction
and the city acceleration is in
and you have the wide
another words
we have no the three-dimensional motion
we have cuts
into
one-dimensional motions
this is a one-dimensional motion
the behavioral only along x-axis and
they did the behavior only along the
white axis
and it is the behavior only along z axis
and if three together
makeup
the actual motion of that particle
won every game go it looks like
this looks like a mathematical zoom
you would say well
then what is going to be like is going
to be phil
not quite
in fact is going to help you the great
deals
first of all if i throw up a time as
bowling class like this
then the whole trajectory
the whole projector isn't long plane in
the political plane
so even though it is in three dimensions
we can always represented
by two x's by two-dimensional at the
white exit and x-axis
soul ready
the three-dimensional problem often
becomes a two-dimensional problem
we will be great success
analyze
these trajectories
by decomposing
is very complicated motion imagine what
it incredibly complicated archivist
and yet we're going to decompose it's
into a motion in the exception
regulators alive of his own independent
of the motion in the uae dot action
which leaves a lot of its own
and of course you always have to combine
the two to know
what the particle
we know they equations so well
from our last lecture
form on the nation will
motion
with constant acceleration
the first line tells you what the
exposition is as a function of time
the index t
tells you that it changing respond
it is the position of people zero
plus the velocity of people zero consti
was one-half
any excuse korea if that is an
acceleration in the eccentric
the velocity you need a comes from
taking did irrevocable dysfunction any
acceleration comes from taking the
derivative of this function
now if we have to
emotion
which is more complicated
which reaches out to two or three
dimensions we can decompose emotion
three
perpendicular axes
and you can't replace every axia by a
wife
which gives you the entire
behavior in the white out action
and if you want to know the behavior in
the direction
you replace every x here by using
and then
have decomposed emotion
in three
directions each obon are
and that's what i wanted to know
i'm going to
throw up
an object
golf ball
or an apple
in twenty six one hundred
and we know that it's in a vertical
plane so we have we only deal with a
two-dimensional problem
this being
a call this mike
x-axis
and i'm going to call this
why why texas
i call is increasing value over x
and i called is increasing value of why
i could have called
this increase in value of life
today i've decided to pull
this increasing value like i'm free in
that choice
i fell upon albrecht
the southern angle
and i see
motion like this morning and it comes
back
for the ground
my initial
speed
right through it
was the zero
and the angle here
its all time
x component
of that initial velocity
it's the zero
co-signed all clap
and the white components
equals
the zero
final
so that's good news for us to feel
the eccentric and if the begin velocity
in the white house
a little later in time
that object
is here
appointee
and it is now
the position vector which we have cold
are often
this factory
nafta back to that is moving
from space
at this moment in time
axle field
is here
and at this moment in time
why okie
now you're going to see
for the first time
vicki gained
by the way that we've divided
to exceeds which led
and independent life prospects
i wanna know everything about x
never has to be no
i wanna know
ready for a moment in time
velocity acceleration only an axe
first a one eleven people zero
wealthy people zero i look there
x zero with the i can show that to be
zero
so i can say x zero zero that's my free
choice
now i need you general action will be
the velocity
the velocity of people zero which we
have cold v zero x
is this a loss of t_v_ zero call sign
off on
and it is not going
why if they're not going to change
because
there is
no it affects
so this time here
zero we only have this warm football
moment in time the velocity accel
rection
is the zero
co sign off
and the apex
equal zero
now i want to do the same
in the excel rection
with plenty
well at time t
i looked at the first equation
uh... various
accurate hero
i know the zero x
that is easier of course i know font
so accesskey
v zero co signed off on
constant
but there is no acceleration
for decades
piak softy
the velocity in the eccentric shin any
moment in time
that that that value creation
that instantly you can relax
it is not changing in time
because there is no acceleration
saudi initial velocity a teacher who is
the same
s_g_ seconds later
andia calibration
now
we're going to do this
for the white house
and are you reading to see
gained forty decomposition
in the white election
we change the x by a wide
so we do its first at people zero
will bear
is becomes white zero i called at zero
i can always former origin zero
i get the zero why primes p
well do you know why is this
quantity the zero
signed all talk
the zero
final file
this is the zero signed off on at is the
philosophy of parent ural and this is a
row
and europe
this is early prime zero
what is the acceleration
in the white are action
time zero
what is the acceleration that has to do
was gravity
there's no acceleration in the
expiration
but you better believe it is one of the
white house
only when we do with the white meat
creations
does this acceleration come in
model when we do with the ecstatic
well if we call the acceleration due to
gravity g
equals plus
nine point eight zero and i always
apology
what's will be acceleration in the line
of action given the fact that i called
this increasing value of life
a mine
nine twenty-eight
which i would also say
always coal miners gene because my g is
always positive
sodas minus
tells the story of king equal zero in
the white election
and now we have to complete it
a time t_v_ full steam
a time people speak
we have the first line there
white zero a zero
so we have why it is a function of crime
one zero zero
so we don't have to
workers that where's mine
because europe
so i get the
zero white pine street
so i get to the zero
final thought
times humko
plus one hop
but it is minus
one-half
gps squared
and now i get
the velocity in the wider action akan
cheated my second line
that is going to be
the hero
final product
minus
vw gti
andy acceleration in the white election
any moment in time
mime-version
and now i have done all i can
who completely decompose
is complicated motion
into to entirely independent one
dimensional motions
next lecture
we're going to use this again and again
and again
and again the collectors not over yet
but i wanted to know that because what
we're going to apply
for many doctors to come
the decomposition
off today
complicated
trajectory into through simple ones
now when you look at this
there's something quite remarkable
and the remarkable thing is
that the velocity accel rection
throughout this whole trajectory if
there is no way a direct if there is no
friction
is not changing
is only the velocity of the wider
election that is changing
it means a biker what these bulbule
i probably did
and it has a certain component index
temecula certain velocity if i lose
myself with exactly that same velocity
is exactly the same way resolvable
orthopedic i could catch the board it
moot it would have to come back exactly
in my hands
that needs to be called a resolving the
calorie shin
in the white election
but the motion in the white election is
completely independent of the expansion
the excel action
doesn't even know what's going on with
the white house reaction
in the actual rection
if i throw an object like this
the eccentric shin
simply very boring really
moves with constant
velocity there is no
dependents
and the wider action on its own
does its own thing it goes up
comes to a halt
and stop
and of course the actual motion is the
samba superposition
of the two
we have tried to find a way
demonstrate this
quite bizarre
behavior which is not so intuitive
that the expelled extreme really live
the life of its own
and the way
re want to do that
is
as follows
we have here is a
bowler
done we can
showed up the global
and we do that in such a way
that the
gold bull
if we do it correctly
exactly comes back here
that's not easy
that there's always and always over just
months
global
and comes back here
not here
not here
not they have f_e_c_ speech would double
inlet an angle in the global will come
back here
once we have achieved that that the cold
war will come back there
then i'm going to give this car
a push
and the moment that it passes for the
switch
the gold ball of fire
bold bold bold will go straight up
a scene from the card
but it hasn't horizontal velocity
which is exactly the same ozone
vocalocity as the caller
for the car
i like my hands as the cold war goes
like this
the prostate always exactly on the golf
balls
always exactly on the table or
and it will work too well
the ball and up
exactly the card
let me first
show you otherwise if that doesn't work
of course it's all over
it recently bought read up
that it comes back to your doesn't do
that i'd or even have to try this
more complicated experiment
so here's the global
even if i have to go now
reasonably close
well hindi film media controls
do
perhaps it would help if we did will be
it
leeway
there goes the gun
you're comfortable
disease just in case
down
as i'm going to push this now
that approach
the gun
will be trinkets
when the middle of the car is here
you've seen how high double dose of ipl
will go
depending upon
how hard i push it they may need here
when they meet there
ready for this
david's work
see you wednesday
