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Now to introduce coordinates let me start
with a simplest case of one-dimensional coordinate
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systems okay Let's assume that i have an infinitely
long parallel road okay which is running along
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this way and there are houses situated along
the road okay
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00:00:30,860 --> 00:00:31,860
.
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00:00:31,860 --> 00:00:35,570
You can imagine a highway or something where
the road is long and there are houses situated
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00:00:35,570 --> 00:00:41,989
along one edge of the road okay Now any road
could be taken as a reference because this
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00:00:41,989 --> 00:00:46,460
is an infinitely long road there are an infinite
number of houses and you can take any house
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as a reference but it might be very easy for
me to take my house as a reference okay So
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i will take my house as a reference and i
call my house as house number 0 okay
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00:00:57,350 --> 00:01:02,560
Of course i could have called this house as
150 or 200 it does not matter but 0 seems
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to be giving you a nice intuitive feeling
that this is the origin this is the reference
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So the reference could be set to 0 and i take
my house number to 0 okay Now what i start
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00:01:15,780 --> 00:01:21,720
doing is there are different houses along
the road i start labeling all these houses
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right So i label the houses 1 2 3 4 5 for
example my friend's house happens to be 150th
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00:01:28,500 --> 00:01:30,740
house from my house okay
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00:01:30,740 --> 00:01:35,500
So this friend's house is 150th house from
the reference house which is my house So i
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00:01:35,500 --> 00:01:41,320
number them as 150 Then i have 151 152 and
so on and it will continue like that There
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are houses to the left of my house as well
So i will label them also 1 2 3 4 5 Now if
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00:01:47,740 --> 00:01:52,729
i say house number 5 you would be confused
Why would you be confused because i am not
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00:01:52,729 --> 00:01:58,110
specifying whether this house number 5 is
to the right of my house or to the left of
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00:01:58,110 --> 00:01:59,780
my house right
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00:01:59,780 --> 00:02:04,759
So once i fix the reference i also need to
specify the direction in which i have to find
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this house So for example we have a mutual
friend and i want to direct that mutual friend
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00:02:10,310 --> 00:02:15,670
to one of my other friend's house right to
my friend's house which is at 150 to the right
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00:02:15,670 --> 00:02:23,920
of my house I have to specify go along right
of my house and stop at 150th house okay
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00:02:23,920 --> 00:02:30,349
So i have to specify both direction as well
as the distance or the house number which
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00:02:30,349 --> 00:02:37,810
i have to go So this is a one-dimensional
system Now we can mathematically represent
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them that is we can convert this housing analogy
to a coordinate system by drawing a line and
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00:02:45,310 --> 00:02:52,090
we label this line as x axis and we call this
point say my house as a reference house with
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00:02:52,090 --> 00:02:53,400
a number 0
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I associate my house with 0 and then i start
numbering all the other houses the house to
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00:02:59,709 --> 00:03:04,510
distinguish between the house to the right
and houses to the left i can say the houses
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00:03:04,510 --> 00:03:10,900
to the right are positively numbered and houses
to the left are negatively numbered So the
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00:03:10,900 --> 00:03:15,989
house number one which was to the left of
my house has now become minus one now i do
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00:03:15,989 --> 00:03:19,170
not have to specify you know go to the right
or go to the left
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00:03:19,170 --> 00:03:24,870
I just have to specify minus 150th house or
plus 150th house may be there is a house in
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00:03:24,870 --> 00:03:31,020
between 150 and 151 this house could be 1505
So i could say “hey go to the location of
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00:03:31,020 --> 00:03:41,700
1505” May be there is a landmark at 2035
or 2038 right So i could then say go to the
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00:03:41,700 --> 00:03:50,120
landmark 2038 or go to the landmark at minus
1536 right So i can specify that and x is
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00:03:50,120 --> 00:03:53,220
the axis that i would be calling okay
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00:03:53,220 --> 00:04:00,530
So what this house analogy has told you is
that i can associate on this road any point
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00:04:00,530 --> 00:04:07,710
right i can specify that point by giving it
a number So i have converted or i have found
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00:04:07,710 --> 00:04:15,810
the method in which i can locate a point in
space by a number I can specify that point
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00:04:15,810 --> 00:04:21,349
in space by a number For example the position
of the nth house can be specified by giving
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00:04:21,349 --> 00:04:28,190
a number n Now let us say i have a friend
okay who does not agree to consider my house
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00:04:28,190 --> 00:04:31,830
as a reference i mean everyone would like
to consider their house as the reference
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00:04:31,830 --> 00:04:37,259
So i have a friend who wants to consider his
house which happens to be house number 1 as
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the reference So what he or she does is that
they keep this house 1 as the reference house
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00:04:44,189 --> 00:04:50,150
okay So in their coordinate system or in their
method of measuring the locations or specifying
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00:04:50,150 --> 00:04:56,129
the locations of houses one becomes 0 n becomes
n minus one
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00:04:56,129 --> 00:05:01,490
Does that mean that the nth house has actually
physically changed No the house is situated
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00:05:01,490 --> 00:05:08,159
where it was it's number that we are specifying
has become different because in my mind 0
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00:05:08,159 --> 00:05:15,249
is the reference house and for his coordinate
system 1 is the reference or 1 is the 0 for
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00:05:15,249 --> 00:05:20,020
that one So because the origins are different
the number that we come up with are different
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00:05:20,020 --> 00:05:22,360
but physically this is the same house
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00:05:22,360 --> 00:05:27,409
The house has not changed just because i have
decided or my friend decided that the origin
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00:05:27,409 --> 00:05:32,059
needs to be different okay So i hope you get
the difference between specifying a coordinate
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00:05:32,059 --> 00:05:38,249
system and specifying a number okay This is
closely related to the fact that vectors we
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00:05:38,249 --> 00:05:42,119
will soon see how to connect vectors here
So this is closely related to the fact that
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00:05:42,119 --> 00:05:47,839
vectors are independent whereas the number
that we associate with a vector is dependent
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00:05:47,839 --> 00:05:53,479
on the coordinate system that i choose
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00:05:53,479 --> 00:05:57,749
So now how do we associate vectors to this
coordinate system we have already established
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00:05:57,749 --> 00:06:03,960
a coordinate system okay and let's say the
coordinate system that i have established
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should allow me to measure distances
.
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So i do not want to say the position of 150th
house i want to say “hey my friend's house
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00:06:11,270 --> 00:06:18,159
which is at 150 is actually at 150 kilometers”
or it is at 150 meters or it is at 150 centimeters
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00:06:18,159 --> 00:06:25,669
okay So there is a landmark at 2038 kilometers
so i want to specify distances out there but
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00:06:25,669 --> 00:06:32,500
you will see what i did over here i had a
distance of kilometer meter centimeter How
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00:06:32,500 --> 00:06:39,319
was this distance or how was this unit of
measure formulated Let's lookay at this I
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00:06:39,319 --> 00:06:44,460
have all these houses it is again see that
i am using my coordinate systems
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00:06:44,460 --> 00:06:50,710
So i can use my coordinate system and say
i measure the distance from my house to my
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00:06:50,710 --> 00:06:57,080
next house along the right which is say 1
and i call this as the unit distance So all
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00:06:57,080 --> 00:07:03,759
other distances along x axis are now measured
in units of this distance okay So this is
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00:07:03,759 --> 00:07:12,249
1 unit so 150th house will be 150 units away
from 0 again there is no specific reason why
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00:07:12,249 --> 00:07:15,749
we have to specify this as 1 unit distance
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00:07:15,749 --> 00:07:22,399
I could specify the distance between midway
from my house to my friend's house or to my
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00:07:22,399 --> 00:07:28,729
neighbor's house as 1 unit then the 150th
house which is my friend's house will be at
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00:07:28,729 --> 00:07:37,539
a distance of 150 into two which is 300 units
if i half the unit So it is clearly my control
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00:07:37,539 --> 00:07:42,409
as to what i should call as a unit and we
pick whatever the unit that would correspond
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to us and we just stick to that unit and this
kilometer meter centimeter are all the choices
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00:07:48,339 --> 00:07:49,339
that we make
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00:07:49,339 --> 00:07:53,919
So if i take 1 kilometer as the unit then
all the other distances can be measured and
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00:07:53,919 --> 00:08:00,169
will be measured as some multiple of kilometer
That multiple need not be integral it need
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not always be 1 kilometer 2 kilometer 3 kilometer
Our landmark could be at 23 kilometers okay
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So i hope that distinction is very well understood
by you
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00:08:12,619 --> 00:08:19,199
Now this is a coordinate system along to the
right there is x axis i have also obtained
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00:08:19,199 --> 00:08:25,789
the units on which i am going to measure now
how do i define a vector Consider the nth
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00:08:25,789 --> 00:08:33,969
house okay I can specify the nth house by
giving its distance from the origin origin
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being my house which i am taking as the origin
or the reference i can give the position of
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00:08:39,430 --> 00:08:50,670
this nth house as n units away from 0 I can
equivalently draw a vector whose length is
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00:08:50,670 --> 00:08:57,800
n and whose origin is at 0 and which terminates
at n
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00:08:57,800 --> 00:09:02,379
Now you can imagine that i am first removing
the coordinate system i just have a vector
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which is lying horizontally of this length
0 to n okay I mean of length this line and
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00:09:09,110 --> 00:09:14,470
now if i want to attach a coordinate system
i can simply imagine a coordinate system over
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00:09:14,470 --> 00:09:19,850
here in which the origin coincides with the
tail of the blue vector okay Which is what
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00:09:19,850 --> 00:09:28,470
the vector n is okay So this vector n can
be represented as n times x hat
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00:09:28,470 --> 00:09:35,790
What is this funny lookaying x hat over here
X hat tells me a unit vector along the direction
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00:09:35,790 --> 00:09:43,060
of x What is a unit vector It is a vector
which has unit magnitude and points in the
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00:09:43,060 --> 00:09:46,899
direction of the coordinate system Here i
have a one-dimensional coordinate system along
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x so this is pointing along x direction has
a magnitude of one so this unit vector have
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indicated by this orange colored vector okay
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So orange colored line arrow is a unit vector
and if i take a unit vector and multiply it
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by a number n then i get the vector n which
has the magnitude of n and it will not be
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pointing along the x direction okay So now
i have shown you two things one there is a
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position which is the location of the space
along the x axis that can be specified by
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giving it a number
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So if you imagine that there is no vector
over here if you imagine that there is no
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vector here but just a number n that is sitting
here on the x axis then the location of this
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00:10:30,310 --> 00:10:37,000
point where n is sitting is nm okay You can
have another position which would be m which
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would be along the x axis Now i have a vector
with no specific reference to the coordinate
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00:10:43,440 --> 00:10:48,870
system However i can introduce a coordinate
system and say that this vector can be represented
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00:10:48,870 --> 00:10:55,370
as n x hat where n represents the magnitude
of the vector and x hat represents the direction
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of the unit vector
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So in this case the unit vectors can only
be directed along x Now suppose i want to
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represent a vector which has same magnitude
n but it will be pointing in the negative
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x direction how would i represent that All
you have to do is you have to represent this
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as minus n x hat You could of course take
a unit vector which will be pointing in the
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00:11:18,189 --> 00:11:22,509
negative x direction you can call that as
some minus x hat
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Then i can multiply that by the required magnitude
say the value of n and then obtain minus n
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x hat What we preferred to do is that we keep
the unit vectors fixed If i want to refer
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to a negative direction i simply multiply
that one by a sign number if the sign number
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is positive it will be pointing in the same
direction as the unit vector if the sign number
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is negative right the multiplier is negative
then it will be pointing in the opposite direction
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okay
.
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So this was the one-dimensional coordinate
system a vector n So let me recap this a vector
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n which is starting at origin o and ending
at a point n is given by n x hat Please note
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00:12:04,040 --> 00:12:07,550
the notation that we are going to use This
will be the notation that we will be using
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throughout our study okay Here the number
n which can be positive or negative okay If
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00:12:14,290 --> 00:12:18,879
we associate and sign along with this one
if i do not have then n is a positive number
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So n is a magnitude of the vector n and x
is the unit vector along x axis If i want
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00:12:24,700 --> 00:12:29,560
to represent a vector which is acting along
minus x direction or which is along the negative
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00:12:29,560 --> 00:12:35,759
x direction then i have to assign sign here
okay I have to say minus n x hat which will
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00:12:35,759 --> 00:12:40,620
then point to a vector whose magnitude is
n that is whose length is n but it will be
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00:12:40,620 --> 00:12:45,110
acting in the minus x direction negative x
direction okay
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00:12:45,110 --> 00:12:50,610
If i take a unit vector and then multiply
it by any number i am essentially stretching
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00:12:50,610 --> 00:12:56,069
the unit vector okay If the multiplier is
less than 1 then i am compressing the vector
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If the multiplier is equal to minus 1 then
i am taking the vector and changing the direction
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00:13:00,319 --> 00:13:07,550
from plus to minus okay Now let us go to two-dimensional
coordinate system Whatever fun we had to have
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00:13:07,550 --> 00:13:13,920
with one-dimensions is all over now let us
go to two-dimensional scenarios okay Now here
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00:13:13,920 --> 00:13:19,799
again the notion of coordinate system as well
as the vector will be different okay
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00:13:19,799 --> 00:13:20,799
.
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00:13:20,799 --> 00:13:24,639
So you can have a vector which is independent
of the coordinate system and then you choose
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00:13:24,639 --> 00:13:29,550
a coordinate system such that the numbers
that you get will be unambiguously defining
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00:13:29,550 --> 00:13:34,670
a vector and it will be simpler to work with
So you have an unambiguous definition of a
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00:13:34,670 --> 00:13:41,079
vector in the chosen coordinate system as
well as an easier method of handling the numbers
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00:13:41,079 --> 00:13:47,060
okay Just to bring out the distinction lookay
at this figure you have a vector which i am
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00:13:47,060 --> 00:13:52,089
calling as vector a which is given by red
colored arrow over here
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00:13:52,089 --> 00:13:57,629
This red colored arrow as in the previous
vectors will have 2 points there is a starting
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00:13:57,629 --> 00:14:02,879
or the initial points Because this is a two-dimensional
coordinate system you might expect that every
150
00:14:02,879 --> 00:14:09,269
point in the position will be specified positioning
space will be specified by 2 numbers So this
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00:14:09,269 --> 00:14:17,050
origin point of the vector a will be specified
by 2 numbers x i and y i Similarly the end
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00:14:17,050 --> 00:14:23,839
point of the vector will be specified by 2
numbers which are x f and y f okay
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00:14:23,839 --> 00:14:29,980
What are these x and y These are the 2 lines
or the directions which are needed to completely
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00:14:29,980 --> 00:14:35,480
specify any point in the coordinate system
In the two-dimensional system i have to specify
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00:14:35,480 --> 00:14:42,589
the location of any point by 2 numbers One
along x and one along y Imagine that i have
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00:14:42,589 --> 00:14:45,870
this plane and then there are houses scattered
all over
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00:14:45,870 --> 00:14:49,959
Now if i want to reach a house which is at
some particular point in this space over here
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00:14:49,959 --> 00:14:56,580
i can say “hey go along horizontally 10
units of houses” or "10th house and from
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00:14:56,580 --> 00:15:05,170
that house go vertically to 20th house to
reach the house which is numbered 10 20" okay
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00:15:05,170 --> 00:15:11,160
This is the meaning of giving 2 coordinates
and you can notice that the directions horizontal
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00:15:11,160 --> 00:15:13,220
and vertical are perpendicular to each other
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00:15:13,220 --> 00:15:17,440
We will talk a little bit about perpendicular
the requirement of perpendicularity later
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00:15:17,440 --> 00:15:22,899
okay At any point we can be considered as
origin so we consider this point o as the
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00:15:22,899 --> 00:15:28,709
origin here So anything to the increasing
a side of x will be positive this would be
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00:15:28,709 --> 00:15:35,500
positive if you go below this line you will
be going minus y if you go to the left you
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00:15:35,500 --> 00:15:41,079
will be moving along minus x direction okay
So any point in this two-dimensional space
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or a plane can be specified by 2 numbers okay
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00:15:46,279 --> 00:15:51,829
So here is a vector a with its initial and
final points Here is a coordinate system that
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00:15:51,829 --> 00:15:57,230
my friend has come up with Now every time
i specify the vector a i have to specify both
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00:15:57,230 --> 00:16:02,519
the initial as well as the final points Now
this might become little tricky okay because
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00:16:02,519 --> 00:16:07,139
every vector needs to be specified by 4 numbers
now one for the starting point and one for
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00:16:07,139 --> 00:16:14,579
the end point On the other hand it would be
wonderful if i take the same origin okay for
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00:16:14,579 --> 00:16:16,389
all the vectors
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00:16:16,389 --> 00:16:23,040
What am i getting by doing so I am eliminating
this need of specifying this x i and y i all
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00:16:23,040 --> 00:16:29,089
the time I do not have to specify x i and
y i all the time If i choose my coordinate
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00:16:29,089 --> 00:16:35,831
system as coinciding with the tail or the
origin of this vector but the coordinate system
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00:16:35,831 --> 00:16:40,680
does not seem to be coinciding What is the
solution I cannot move the vector of course
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00:16:40,680 --> 00:16:45,560
i can move the vector but normally what we
think of is we do not move the vector we move
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00:16:45,560 --> 00:16:46,660
the coordinate system
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00:16:46,660 --> 00:16:50,779
There are 2 equivalent ways of doing the same
thing Either you move the vector parallelly
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00:16:50,779 --> 00:16:56,410
until you reach the origin of the coordinate
system okay We can do that one or we simply
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00:16:56,410 --> 00:17:01,449
move the coordinate system until you reach
the origin and moving the coordinate system
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00:17:01,449 --> 00:17:06,160
not only means parallelly moving them you
can also twist and rotate the coordinate systems
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00:17:06,160 --> 00:17:09,449
okay Some of these things will become important
later
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00:17:09,449 --> 00:17:13,930
At this point let us not clutter too much
about that so what i am going to do is that
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00:17:13,930 --> 00:17:21,130
i am going to redraw my lines y and x such
that the origin of the coordinate system o
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00:17:21,130 --> 00:17:27,370
right coincides with the origin of the vector
a Thus a vector a will now be specified only
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00:17:27,370 --> 00:17:35,659
by 2 points x and y Why because this 0 and
0 the origin is implicitly understood If i
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00:17:35,659 --> 00:17:42,149
had another vector i could draw a line from
the origin to the other point and that other
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00:17:42,149 --> 00:17:45,730
vector could also be specified by only 2 numbers
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00:17:45,730 --> 00:17:51,429
This is the advantage of coinciding or making
your origin of the coordinate system coincide
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00:17:51,429 --> 00:17:57,130
with the tail of all the vectors How about
the vector difference Well you have to wait
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00:17:57,130 --> 00:18:03,220
for a few slides to see how to define the
sum and addition of vectors when you have
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00:18:03,220 --> 00:18:10,049
2 different vectors here okay But for now
please remember that you can draw a vector
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00:18:10,049 --> 00:18:15,650
from the origin to any other point in the
two-dimensional space All vectors are referred
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00:18:15,650 --> 00:18:19,100
with respect to origin okay
.
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00:18:19,100 --> 00:18:24,399
So you have a vector a which is referenced
with the origin o here and let's say this
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00:18:24,399 --> 00:18:32,840
vector a is terminating at a point 25 and
3 What is this 25 and 3 mean If you forget
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00:18:32,840 --> 00:18:39,460
the vector a this simply gives you the location
of the point 25 and 3 in the two-dimensional
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00:18:39,460 --> 00:18:46,240
space that is described by this vector x and
y correct So i have a vector x and y and all
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00:18:46,240 --> 00:18:49,990
points are specified by 2 coordinates
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00:18:49,990 --> 00:18:58,809
So in this particular case the vector a goes
from the origin and terminates at 25 and 3
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Now to go from o to a i can of course go along
the red line or i can go along the horizontal
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direction until i reach this point You can
see that where i am reaching this one and
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then continue in the vertical direction
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Now if you remember how we added two vectors
you had one vector tail origin to that you
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added another vector that is you brought in
another vector by parallelly translating it
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such that the tail of the second vector coincided
with the head of the first vector Now i have
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a tail of this vector which is now getting
added to the vector which is horizontal So
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i have a horizontal vector then a vertical
vector
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The sum of these 2 vectors is obviously the
vector o a which you can see very clearly
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over here What are the 2 vectors that i am
adding The horizontal vector which is given
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by this blue line and then a vertical vector
which is given by this blue line What is the
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magnitude of the horizontal vector It is exactly
this 25 and in which direction does it point
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It points along the x direction So i can represent
this vector itself as 25 x hat right This
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vector is 25 x hat and this is precisely the
vector that is giving you this horizontal
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00:20:19,919 --> 00:20:20,919
vector
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00:20:20,919 --> 00:20:26,279
So this horizontal vector is characterized
by x hat a x where x hat is the direction
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of the vector which is along the x axis and
a x is the magnitude of that vector that is
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of the horizontal vector which is 25 so this
is another way of saying that you move 25
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00:20:38,750 --> 00:20:44,790
units along x and then move 3 units along
y and if you move along y you are actually
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creating a vector here which again is nothing
but a parallelly transmitted vector y hat
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a y and what is a y here A y is 3
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00:20:55,179 --> 00:21:02,500
So this original vector o a has been resolved
or decomposed into 2 vectors the sum of 2
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vectors both this vectors are so one of the
vectors is along x the other vector is along
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y These two themselves are perpendicular to
each other You can see that from the graph
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the vector a x x hat is perpendicular to the
vector a y y hat okay Similarly as we defined
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a unit vector for the x axis i can define
a unit vector for the y axis and i have actually
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used that definition of unit vector of the
y axis when i am writing this vector y hat
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a y okay
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00:21:33,520 --> 00:21:39,800
So the vector o a has been resolved into 2
components that is sum of two vectors one
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vector along x and the other vector along
y So i have two vectors one along x and one
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along y The sum of these two is giving me
the vector o a okay Keep this in mind okay
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00:21:52,580 --> 00:21:53,580
.
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00:21:53,580 --> 00:21:57,390
Now we move from two-dimensional coordinate
system to three-dimensional coordinate system
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Now this is why electromagnetic is sometimes
thought to be very abstract mathematical and
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at of subject because you have to work with
three-dimensions However your work will be
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simplified if you understand the coordinate
systems and if you choose an appropriate coordinate
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system for your problem okay
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We will see the tragic consequences of choosing
a bad coordinate system to solve a particular
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electromagnetic problem sometime later For
now let us try to understand the three-dimensional
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coordinate system If you have followed the
discussion of one-dimension and two-dimensional
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coordinate system extending this to three-dimensions
should not be a problem except that it will
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be little mentally taxing in visualizing the
three-dimensional vector
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If you go back to two-dimensional case you
had two lines may be we can take this as the
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00:22:46,070 --> 00:22:50,769
two-dimensional coordinate system example
so you had 2 lines which were perpendicular
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to each other and these 2 lines are mutually
perpendicular to each other and you define
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a unit vector along one and you define another
unit vector along another axis So you have
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a unit vector x hat you have a unit vector
y hat okay
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00:23:06,159 --> 00:23:12,080
So any point was actually given by intersection
of 2 lines For example you lookay at this
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point 25 3 this point at which the vector
a is terminating is actually the intersection
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of 2 lines This line this horizontal line
that is shown here is the horizontal line
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which corresponds to the constant value of
x okay So this line is x is equal to 25 Whereas
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00:23:32,769 --> 00:23:38,809
this horizontal line that i have shown here
dash is the line y is equal to 3 okay
255
00:23:38,809 --> 00:23:45,010
This line is x is equal to 25 you can actually
stretch this along this direction and no matter
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00:23:45,010 --> 00:23:49,669
where you are on this point the value of x
will always be the same The value of x along
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this dashed vertical line will always be same
and it will be equal to 25 The value of y
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along this horizontal line will be the same
and it will be equal to 3 Only thing is x
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00:24:02,230 --> 00:24:07,850
will be changing in the horizontal line whereas
x will be constant here and y will be changing
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as you move up and down okay
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So any point in two-dimensional coordinate
system was represented as intersection of
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2 points Now on a three-dimensional coordinate
system you are lookaying at intersection of
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mutually perpendicular axes which form 3 planes
okay What is this plane See now lookay at
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this x is equal to 0 plane i have 3 mutually
perpendicular axes that means that axis x
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will be perpendicular to axis y axis y will
be perpendicular to axis z okay Of course
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axis z is perpendicular to axis x Lookay at
x is equal to 0 plane okay
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00:24:49,909 --> 00:24:56,090
All points on this plane have the x value
which is equal to 0 and if you lookay at this
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horizontal plane this is z is equal to 0 plane
because all points on this plane have a constant
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value of z which is 0 in this case Similarly
you have another plane which is y is equal
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00:25:10,500 --> 00:25:16,890
to 0 you can go up which means z is changing
you can go along the direction of x which
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00:25:16,890 --> 00:25:22,260
means x is changing but you are not moving
away from the plane which means y is constant
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00:25:22,260 --> 00:25:27,000
So you have y is equal to 0 or y is equal
to constant In this case the constant is 0
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plane You have x is equal to 0 plane you have
z is equal to 0 plane and the intersection
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point of all the 3 planes defines the origin
Now if i want to specify a general x y z point
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how do i specify Well you have to put up 3
mutually perpendicular planes there you have
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00:25:43,630 --> 00:25:49,210
to put up X is equal to constant plane y is
equal to constant plane and z is equal to
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00:25:49,210 --> 00:25:50,210
constant plane
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00:25:50,210 --> 00:25:56,480
So you have to put up 3 mutually perpendicular
axes composed of 3 different planes so as
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to specify any point So clearly any point
in a three-dimensional space will have to
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00:26:01,530 --> 00:26:09,130
be specified by 3 points okay Note the direction
of the vector arrows over here i have a vector
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00:26:09,130 --> 00:26:13,110
x which is increasing in this direction i
have a vector y which is increasing in this
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00:26:13,110 --> 00:26:16,880
direction and i have a vector z which is increasing
in this direction okay
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00:26:16,880 --> 00:26:24,220
The specific directions were chosen to satisfy
what is called as right-handed coordinate
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00:26:24,220 --> 00:26:29,019
system such that the resultant coordinate
system is right-handed coordinate system What
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00:26:29,019 --> 00:26:33,680
is right-handed coordinate system Right-handed
coordinate system is shown here You imagine
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00:26:33,680 --> 00:26:39,620
yourself as using your thumb finger to point
along the z direction okay
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00:26:39,620 --> 00:26:45,460
So if you imagine yourself as using the thumbed
point along the z direction you will see that
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00:26:45,460 --> 00:26:50,360
the index finger will be pointing along the
x and the curved fingers will be pointing
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00:26:50,360 --> 00:26:56,880
along y direction okay So you have thumb pointing
along z direction and the index finger pointing
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00:26:56,880 --> 00:27:02,460
along the x direction and the curling is pointing
along the y direction The curled finger would
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00:27:02,460 --> 00:27:04,260
be pointing along the y direction
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00:27:04,260 --> 00:27:10,679
So in another words if you go from x to y
you will be going up along the z direction
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00:27:10,679 --> 00:27:15,299
You can imagine a different coordinate system
in which if you go from x to y you will be
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00:27:15,299 --> 00:27:21,519
moving along the minus z direction okay Minus
z with respect to this particular z i am talking
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00:27:21,519 --> 00:27:26,309
about that would be a coordinate system that
would be lookaying like this okay It would
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00:27:26,309 --> 00:27:30,830
be an inverted coordinate system and sometimes
called as the left-handed coordinate system
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00:27:30,830 --> 00:27:31,830
.
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00:27:31,830 --> 00:27:35,470
Let us lookay at how to specify the points
on a three-dimensional coordinate system To
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00:27:35,470 --> 00:27:42,250
specify a point which is x y and z okay i
need 3 planes as i have just told you i need
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00:27:42,250 --> 00:27:50,529
3 planes So if you lookay at this plane this
plane should be erected at x is equal to some
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00:27:50,529 --> 00:27:57,289
constant okay whatever the value that is given
to me and this plane you can see here is to
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00:27:57,289 --> 00:28:02,840
be erected at y is equal to constant So whatever
the value of y that i need to specify i have
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00:28:02,840 --> 00:28:05,509
to put up a plane here
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00:28:05,509 --> 00:28:09,690
The intersection of the planes x is equal
to constant and y is equal to constant will
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00:28:09,690 --> 00:28:15,990
give me 2 coordinates x and y To get the third
coordinate i have to now put up a plane which
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00:28:15,990 --> 00:28:22,110
is z is equal to constant okay So 2 planes
will give me 2 coordinates the third plane
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00:28:22,110 --> 00:28:27,910
will determine the final point or the final
coordinate of the location in space Again
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00:28:27,910 --> 00:28:33,090
you have three mutually perpendicular directions
which is one vector pointing along x one pointing
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00:28:33,090 --> 00:28:36,370
along y and one pointing along z direction
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00:28:36,370 --> 00:28:43,730
For example consider point p and q point p
is -2 2 and 3 Which means that i have to put
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00:28:43,730 --> 00:28:50,220
up a plane at x is equal to -2 x is equal
to -2 occurs beyond this For example this
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00:28:50,220 --> 00:28:56,880
is the x is equal to -2 and then i have to
move 2 point 2 on y axis so i can actually
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00:28:56,880 --> 00:29:03,549
move along the x line to the y axis to get
me this point okay This is equivalent of putting
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00:29:03,549 --> 00:29:08,870
up a plane at y is equal to 2 and plane at
x is equal to -2 the intersection of those
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00:29:08,870 --> 00:29:11,529
two will be at -2 and 2
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00:29:11,529 --> 00:29:18,490
Now if you move up because this is 3 if you
move up to the point 3 or the plane where
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00:29:18,490 --> 00:29:25,899
z is equal to 3 you will end up with point
p So to get to point p you move -2 along x
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00:29:25,899 --> 00:29:34,950
2 along y and 3 units along z Similarly to
get to y which is 3 -2 and 2 you have to first
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00:29:34,950 --> 00:29:39,470
move along x is equal to 3 that is to move
to the plane x is equal to 3 So i moved to
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00:29:39,470 --> 00:29:41,460
this plane x is equal to 3
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00:29:41,460 --> 00:29:46,820
On this plane y and z values could be anything
but y is equal to 2 is given So i need to
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00:29:46,820 --> 00:29:51,600
now move to y is equal to minus 2 planes So
i need to setup y is equal to minus 2 plane
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00:29:51,600 --> 00:29:57,389
Now when i bring these 2 planes together i
am at a point which is 3 and 2 Now i have
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00:29:57,389 --> 00:30:03,639
to move to point 2 along z so which means
i have to erect a plane at z is equal to 2
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00:30:03,639 --> 00:30:10,809
So i have moved to point 2 So this q is defined
by 3 numbers 3 - 2 and 2 okay
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00:30:10,809 --> 00:30:16,740
A different notation is to just give you the
vector i mean just give you the point p and
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00:30:16,740 --> 00:30:22,169
give the coordinates of the point So p and
in brackets you give the coordinates say in
328
00:30:22,169 --> 00:30:28,749
this case it is -2 2 and 3 and for q it would
be 3 -2 and 2 okay
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00:30:28,749 --> 00:30:29,749
.
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00:30:29,749 --> 00:30:35,789
So this is how to specify locations on a three-dimensional
coordinate system I hope that you have understood
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00:30:35,789 --> 00:30:43,610
this particular point and i suggest that you
try to graph more number of points to become
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00:30:43,610 --> 00:30:45,659
familiar with this particular coordinate system