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How do I specify points Go back to our idea
of specifying points on a road on a plane
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On a road I show the particular co-ordinate
axis How this were lined up on to the right
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of my reference house or to the left of my
reference house When I graduated to two dimensions
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how this scattered all over the plane and
if I wanted to specify the location of a particular
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house right
.
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So if this is the plane if I wanted to specify
the location of this house I had to specify
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them by giving the x and y co-ordinates I
had to tell you how to move along the x direction
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what value you are going to move and proceed
vertically until you reach this house this
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is XY plane and this particular point was
given the co-ordinates x and y indicating
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the amount that you had to move along x and
the amount that you had to move along y
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So if you wanted to also specify the height
you had another option going to z co-ordinate
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system This is the rectangular Cartesian co-ordinate
system that we studied Is there another way
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of specifying this location of the house answer
is yes Instead of specifying the distances
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to move along X and Y why not I specify the
distance from the reference or the origin
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to the house and the angle of the road assuming
that there is a road in that particular angle
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what is the angled road that I have to take
in order to reach that house
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So all I am saying is that you start from
the origin and you want to reach this point
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you head towards this house at an angle of
say phi as measured from the x axis because
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angle is some quantity that you will always
have to measure with respect to two lines
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So measure angle with respect to x axis this
is a conventional choice and what distance
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you have to move That distance let us call
as r
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Can I now find out what is r and phi Here
is a small point In some of the textbooks
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you will find that this r is actually denoted
by rho I am not using rho for a very important
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reason rho for us is the charge density If
I use rho there might be a case where I will
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be writing rho L rho hat and at least for
me this kind of starts getting confused Rather
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than that I know that r stands for radius
okay
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So I am going to use r as my variable over
here and applying the law of right angle triangle
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if this is x and this y obviously r is equal
to square root of x square plus y square Now
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you see why I used r in the previous example
preciously because I was anticipating that
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I am going to discuss cylindrical co-ordinates
with you that is why I used r is equal to
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square root of x square plus y square How
do I calculate phi Well use trigonometric
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relation
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Phi is nothing but Tan inverse or inverse
tan of y by x
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So these two will give me the equal end points
on the cylindrical co-ordinate system just
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as the x and y points would give me the value
of the point or the co-ordinates of that particular
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point On three dimensional case you have to
imagine that there is a cylinder of appropriate
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radius
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I am still going to use by xyz Cartesian co-ordinate
system but now I am going to imagine a cylinder
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of some radius r and if I want to locate a
point p here all I have to do is to give the
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radius of the cylinder and also give you the
height of the point above the z is equal to
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zero plane So I have this height which is
z okay Then I also have to specify what is
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the angle
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So inorder to do that when I have to drop
down this particular like I have to drop down
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this line on to the xy plane and then measure
the angle of this line with respect to the
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x axis so as to give me the angle value phi
Okay so probably a slightly easier way to
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show you that would be to take up this piece
of paper roll it up Now I have a cylinder
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here This is my z axs imagine that this is
my x and y plane
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I am keeping on this board which is the xy
plane and this is the z axis To specify any
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point here I have to specify at what height
I am and what is the radius of the cylinder
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And if I want to find out the angle I have
to drop this line down to the xy plane and
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then connect the origin to that line inorder
to give me phi So this is the z axis or this
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is the z value that I am going to get
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This is the radial value or this would be
the radial value or if you take the origin
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here that would be the radius value here This
is the z axis this is the r value and then
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you drop down this point on to the xy plane
which would come out to be somewhere over
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here let’s say This axis would then become
the this angle would then be the value of
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phi So quickly let me finish up by giving
you the values the lines along r along phi
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and along z
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Along r is fairly simple this is dr this is
dr in the direction of r in the increasing
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radial direction Along phi interestingly will
have to be r d phi in the direction of phi
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because phi itself is angle it is not distance
you need to measure that and you need to multiply
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with the appropriate radius that you are looking
at So if this is your radius r and this is
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the distance that you have moved d phi then
the arc length here is r d phi and that is
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the line segment here and along z it is simply
dz z hat no change from the rectangular co-ordinate
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system
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So how do we obtain the unique vectors r phi
and z that I have written down for the cylindrical
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co-ordinates in the last lecture Well let
us recap the cylindrical co-ordinate thing
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Now this is a cylindrical co-ordinate system
I am writing everything in this two dimensional
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case that is with only x and y here because
for z it is essentially the same as rectangular
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co-ordinates
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So there is no change in the rectangular co-ordinates
line segment So let us not get bothered about
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the 3 line segment for that one So you have
the x and y co-ordinate system over here Any
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point here we have previously mentioned as
x and y Now we are representing this as a
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point r and phi Okay So where should the r
and phi unit vectors be located Well remember
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what is r r is the distance from the origin
to the circle over here
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So it is increasing in this particular direction
So if I want to write down a circle of a different
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radius I will be writing down a circle like
this So the circle is actually expanding So
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r is increasing in this way So the unit vector
for r would also be along this particular
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direction This point the r will be along this
direction In which direction will unit vector
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for phi be located first of all we know that
this is the angle phi measured with respect
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to the x axis
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So the unit vector must be in the direction
of increasing phi So phi must be along this
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direction So you have r vector and phi vector
I can rewrite these vectors here in slightly
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better way
.
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So I have r vector unit vector and the phi
vector here And the same point I also have
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the unit vector for x and y with the idea
that this angle is phi this angle is 90 minus
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phi this angle is phi So the question is what
is the relationship between the vectors rxy
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and phi What is the relation I know that any
vector in this particular xy plane can be
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expressed in terms of x and y unit vectors
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So I can write down the r vector as something
times x vector plus something else times v
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vector What is that something else You can
see that the relation between r and phi can
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be obtained by taking this r and then if you
imagine that this is one and this angle is
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phi you can find out what is the length of
this particular component and this particular
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component length will be cos phi and this
other length will be sin phi
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So this something times x hat plus something
times y hat in these equation are expressed
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cos phi So I have r is equal to x cos phi
plus y sin phi So I have sin phi Okay What
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is the magnitude of r The magnitude of the
r unit vector should be equal to one and is
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this equal to one Yes because this will be
equal to square root of cos square phi plus
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sin square phi this is equal to one
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Similarly you can show that the unit vector
for phi can be written as because this phi
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is in this particular direction Sorry this
is phi So phi along y will be cos phi along
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x will be pointing in the negative direction
So phi hat vector or the unit vector along
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phi is given by minus sin phi x hat plus cos
phi y hat Okay So I sort of talked about a
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vector along a particular direction
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So I have shown you that r vector can be written
as cos times phi along x and sin phi y and
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phi vector can be written as minus sin phi
x plus cos phi y So I talked about this particular
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thing that r vector along x is cos phi times
x hat r along y is sin phi times y hat What
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exactly did I mean when I said that one vector
is along another vector What did I mean by
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let us say if I take two general vectors let
us call them as vector A and vector B both
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defined in a particular common origin M
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So what do you I mean when I say that vector
B along vector A What do you I mean by that
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What I mean is that in order to tel you what
I mean I need to introduce you to dot product
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Dot product is an example of vector algebra
multiplication of the two vectors We talked
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about two vectors and we added the two vectors
we subtracted the two vectors but we did not
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introduce how to multiply the two vectors
and that is precisely what we are going to
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do now
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We are going to multiply two vectors and the
result of this multiplication is going to
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be a scalar or a number that is why sometimes
dot product is also called as scalar product
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Dot product is also called as scalar product
So what is this dot product Take two vectors
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A and B Now I have two vectors over here This
stick represents a vector This is a good way
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to show the vector and have another vector
B
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Dot product is defined as the length of the
component B along the vector A That is imagine
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that if someone takes up another ruler or
a scale and then drops a perpendicular from
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the tip of the vector B to the vector A to
the point on the vector A So this tip would
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fall here So when this falls you can then
look at the what is the length from origin
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to this one So this particular length gives
me the dot product of the vector A and B
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So what is a dot product of a vector with
itself the length of the vector itself will
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be the dot product because vector dropping
on to itself will be exactly equal to this
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particular length In a general scenario I
have this vector and a vector A and we now
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take a perpendicular and drop it over here
you are going to get the length of the vector
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B along vector A That is I dropped down a
perpendicular from B to A and this length
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that I have is called the dot product of the
two vectors A and B and this is indicated
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by A dot B
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So there is actually a dot here Sometimes
it is very difficult to find the dot So please
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look for the dot here So here is a dot between
the two vectors A and B and the result of
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the dot product will always be a scalar because
it is only giving me the length So this is
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dot product but can I associate a vector with
this length Yes I can because I know what
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is the length
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If I multiply this length by the unit vector
so along the direction of A I will get the
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vector value or the vector component of B
along A So this is exactly what I meant when
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I said I have r hat vector which is the unit
vector in the cylindrical co-ordinate system
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and then I have the x hat vector which is
the unit vector in the Cartesian co-ordinate
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system but I know that both r can be expressed
in terms of x and y
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We have already seen how to decompose any
vector So if I now drop a perpendicular from
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r to x the length of this perpendicular will
be exactly equal to so if this is the angle
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phi this length is cos phi why Because the
length of this is equal to one So if you look
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at this projection this will turn out to be
cos phi How do I associate a vector with this
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00:14:30,300 --> 00:14:34,730
cos phi I need to multiply this one by the
unit vector along x direction so I am going
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to get cos phi x hat
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So more specifically the dot product between
two vectors A and B is given by the magnitude
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of the vector A times the magnitude of the
vector B and the angle between the two vectors
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There are two angles here But we are looking
for the smallest angle between A and B So
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if the angle between the two vectors is theta
AB then the dot product or the length of the
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component of B on A is given by magnitude
of A times magnitude of B
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Sorry these are vectors and cos theta AB and
this will be a scalar quantity and you can
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of course multiply this one by the unit vector
along A to get the vector component of B along
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A We also call this as projection of B on
to A okay Projection of B on to A is given
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by the dot product value which gives me the
length of B along the vector A times the unit
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vector along direction A Can you figure out
what is the projection of A on to B
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Well it might not come as a surprise but the
length of the component of B along A will
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exactly be equal to the length of A long B
right So if B to A that projection value is
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something A to B projection value will also
be the same So this will be you can either
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write this as B dot A or A dot B does not
matter giving us another rule for dot product
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that dot product is commutative A dot B is
equal to B dot A
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So this will be A dot B times vector B okay
So this is the projection of A onto B and
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this is the vector projection of A onto B
So I hope that in the spirit of this discussion
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this relationships of r is equal to cos phi
x hat plus sin phi y hat and phi hat is equal
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to minus sin phi x hat plus cos phi y hat
are very clear to you now Two of other things
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that we have not finished with the cylindrical
coordinates
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.
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How do I write down the vector surface areas
First consider the vector surface area along
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r direction That is I need to hold r constant
and vary the other two variables So how do
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I hold r constant well you go back to this
illustration over here so now imagine that
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I have this particular cylinder of constant
radius r now I have to hold r constant and
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I have to move along two directions along
phi and along z
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So I move along phi moving along phi means
rotating or circulating this particular this
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one So let us say I move along phi by an amount
of d phi How is this an amount of d phi Remember
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from this point you can drop a line to the
origin origin would be somewhere at this particular
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point So you can actually prick some holes
here This will be my y axis and this will
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be my z axis and then there will be another
one which will be x axis
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So if you now look at this point you can draw
a line from this point to the x axis and anther
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point at which point I have ended you can
draw another line The total angle you have
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moved will be d phi and the length that you
have moved will be r times d phi because length
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arc length is not just the angle d phi it
is r times d phi so that length will be r
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d phi How do you move along z
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Let us assume that I am going to move along
z in this direction so I am going to move
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upwards in the z direction how much I have
to move I would be moving a length of dz So
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r d phi dz and I can complete this complete
square or the particular surface element assuming
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all of this to be very small then the total
area that I have generated which would be
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pointing along the constant r direction or
in the radial direction will be given by r
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d phi multiplied by dz okay
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Now suppose I want to find out the surface
area along the phi direction so along phi
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direction means I have to hold this particular
phi to be constant So this is my phi so you
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can imagine that something is coming out like
this so I am holding phi constant over here
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The paper is little thick therefore I am having
some amount of trouble but this is a particular
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phi You can see there is some amount of phi
over here which is constant
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Now in what two directions I can move I can
move along r and I can move along z So I can
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move along r which means that this is going
to be little tricky for me but to move along
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r from a particular point would be to move
like this and then move along z would be to
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move upwards I can complete the square again
or the parallelogram again so that I get the
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component of this vector this is the component
of the vector surface area along the phi direction
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So this pencil or pen indicates the direction
of phi and this vector area will be equal
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to dr dz and it will be pointing in the phi
direction There is no phi here It is just
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dr dr gives me the length along r dz gives
me the length along z How am I going to get
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the surface area along z Well I need to move
along phi and I need to move along r right
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So this particular lock or this particular
plane is the z is equal to constant plane
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If I move along r I am moving along this and
if I am moving along phi I am moving rotating
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around here on a constant r values so I am
moving r that is I am moving dr and I am moving
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r d phi So I have r d phi dr along z direction
So these are the 3 surface vector surface
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elements that you will be meeting later and
which you need to keep in mind
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So hopefully my very crude experimental or
graphical way of showing you how the surface
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elements have worked and you can now very
well understand whatever the surface areas
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The vector surface area along the constant
r is move along phi you are going to move
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r d phi and then move along z you are going
to move dz attach the unit vector along r
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You can also move along constant z you are
going to move along r
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You are going to move along phi and attach
the z unit vector moving along the constant
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phi plane will be equal to move along r and
move along z and attach phi What is the volume
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element Well volume element is how much you
move along r how much you move along phi and
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how much you move along z that could be r
dr d phi dz okay So hopefully this you will
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be familiar Now we said something about the
vectors in cylindrical co-ordinate system
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Remember in the Cartesian co-ordinate system
at any location of the vector the corresponding
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vector was also very easy to find out The
corresponding vector was x x hat plus y hat
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The direction of x and y did not matter where
I was situated right on this two dimensional
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case a plane I don’t care where I am situated
because the x and y direction would always
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point
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For example if this is my two dimensional
plane you have to imagine that my hand fills
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out the entire area the direction of x will
be along this okay My thumb is pointing along
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y So at this point the direction of x is pointing
here The direction of y is pointing upwards
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If I go at this point the direction of x is
pointing here the direction of y is pointing
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here So the unit vectors of x y and in fact
also on z do not change when I move at different
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points
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Is that the same case with the radial vectors
or the vectors in the cylindrical co-ordinate
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system Well we will try okay So how do I specify
the vector in cylindrical co-ordinate well
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I go back and relate them to the Cartesian
co-ordinates R hat is given by cos phi x hat
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plus sin phi y hat and z will be equal to
z itself Phi hat is equal to minus sin phi
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x hat plus cos phi y hat
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Now consider two cases first phi is equal
to zero which means that I am actually along
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x axis There is no angle change on there So
with phi is equal to zero what will happen
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to r hat and phi hat vectors Z will not change
Z will still be pointing upwards It would
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be pointing along z What happens to r cos
phi with phi is equal to zero will be one
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sin phi with phi is equal to zero will be
zero So r hat at phi is equal to zero is pointing
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entirely along the x axis
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So if this is my x axis and this is my y axis
r hat is pointing along this direction interesting
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What happens with r at phi is equal to phi
by 2 90 degrees Go back and substitute over
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here R at phi is equal to phi by 2 will be
cos phi by 2 which is zero sin phi by 2 which
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00:24:10,290 --> 00:24:16,929
is one so you are going to get this fellow
along y hat So the vector at phi is equal
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to pi by 2 the r vector will be pointing along
y direction Sorry this is r
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So at any other point it would be pointing
along a different direction depending on the
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value of phi What it the lesson here The direction
of the vector r depends on phi This is a very
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very important lesson which is not found in
the Cartesian co-ordinate system So you would
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be not really thinking about this one when
you try a different co-ordinate system But
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it is very very important to note that except
for Cartesian co-ordinate system in general
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in other co-ordinate systems
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The direction of the unit vectors do change
as you go at different points in space Okay
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this is very very critical that you remember
this Similarly what will happen to the phi
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vector phi vector at phi is equal to zero
will be pointing along y direction right because
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cos phi will be equal to one at that point
Same phi vector at phi is equal to phi by
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00:25:19,509 --> 00:25:23,809
2 it is interesting that the same vector will
be now pointing along at phi is equal to phi
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00:25:23,809 --> 00:25:27,470
by 2 it would be pointing along minus x hat
direction
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So again the phi vector also depends on phi
So does it mean that I have to all the time
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specify r as a function of phi phi hat as
a function of phi well no Because it will
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simply clutter up our equations We are not
going to write down this functional dependence
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on phi every time but we understand that in
cylindrical co-ordinate system and the next
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00:25:49,809 --> 00:25:54,220
co-ordinate system that we are going to discuss
called spherical co-ordinate system the direction
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00:25:54,220 --> 00:26:01,419
vectors the direction unit vectors are going
to be different at different locations in
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space
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You might ask where is this important Well
this becomes very very important when you
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have vectors inside the integrals Remember
in the last example there was a vector inside
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the integral but the vector was actually constant
I would move up and down along z axis but
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the direction of x or y and decide does not
depend up on where I was located If you try
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doing that same with integral of d5 and you
have r vector in the cylindrical co-ordinate
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system this r will depend on phi
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So different values it will depend on phi
So I do I solve these integrals Turns out
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that these integrals can be solved provided
you convert cylindrical co-ordinate system
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vectors into rectangular co-ordinate systems
So call this as c call this as r you need
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to convert from cylindrical to rectangular
co-ordinate systems And we will look at such
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conversions for a minute now okay
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And in order to find this conversion methods
we are going to use the concept of dot product
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00:27:04,799 --> 00:27:11,309
that we have introduced So go back to this
equation r hat is equal to cos phi x hat plus
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sin phi y hat In this equation if you forget
about y for a minute and look at this equation
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00:27:17,779 --> 00:27:23,739
r hat is equal to something times unit vector
along x Remember the definition for the dot
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00:27:23,739 --> 00:27:30,129
product A dot B is equal to some length times
the vector component of B along A was equal
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00:27:30,129 --> 00:27:36,669
to dot product value times the unit vector
along x correct
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00:27:36,669 --> 00:27:42,229
So that was precisely what we discussed in
terms of dot product So in this case you can
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clearly see that this cos phi would represent
the dot product of the unit vector r hat with
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00:27:49,879 --> 00:27:55,850
the unit vector x hat Similarly sin phi would
represent the dot product of unit vector r
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00:27:55,850 --> 00:28:03,149
hat and the unit vector along y direction
and similarly minus sin phi will be the dot
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00:28:03,149 --> 00:28:12,130
product value along x and cos phi will be
the dot product value along y sorry dot product
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00:28:12,130 --> 00:28:16,259
will be sin phi and cos phi for phi hat and
x and phi hat and y
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00:28:16,259 --> 00:28:22,669
The minus sign is associated with the x direction
That it could be pointing in the x direction
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00:28:22,669 --> 00:28:28,749
So with these knowledge we will now be able
to transform one vector in the cylindrical
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co-ordinate system into vector in the rectangular
co-ordinate system