Mathematics - Multivariable Calculus - Lecture 4

So, we're back to multi-variable calculus, and
before I begin the new topic, I wanted to add a couple of
things to our discussion last Thursday about formulas for the
area in polar coordinates.
Remember, we discussed polar coordinates, and one of the
questions which arises here is to find the area of the
following kind of picture.
You have a curve, which is given by the equation r equals
f of tetha, where r and theta are the two polar coordinates,
which we introduced last time, right.
And, in this polar coordinate's natural bounds for such an area
would not be the vertical lines, like x equal a, x equal
b, the way we used to do it in rectangular coordinates.
That's the old picture.
Now, it's more natural to give bounds in terms
of the angle theta.
So, it's going to look like it a section, like this. between
some angle alpha and beta.
So, we will assume that alpha is less than beta.
And, theta will be between alpha and beta.
And, so we look at this figure which is bounded by the lines,
theta equals alpha, theta equals beta, and the curve.
This is the curve.
And, the question is to find the area of this.
So, I just briefly touched on this, the very end of last
lecture, and I just wanted to tell you how to
get this formula.
Which I wrote down without explaining it last time.
So, the idea is again to break this, to break this picture
into small angles, which would be within some angles, delta,
theta, and say here you'll have r, and just look at the area of
this sector in the disc, right.
So, we approximate now our curve not by horizontal lines
or line segments, but we will now approximate our curve by a
small segment of a circle, because that's more natural,
that's a more natural object in the polar coordinate system.
OK.
So, the question then becomes, what is that area of
this little sector.
In the polar coordinate system, the analogous
picture will be like this.
So, we would have some delta x and delta y, right.
And, so the formula for the area would be just delta
x times delta y, right.
And, more precisely in fact, we don't look at it as a delta y,
but we just look at it as a y-coordinate of a graph, so it
would be more precisely y times delta x.
And, that's what after applying the procedure of breaking into
small pieces give us the formula for the are in terms of
an integral, like this y, dx.
In other words, y delta x, the elementary area gives rise
to this integrand y, dx.
So, we have to approach this problem in
exactly the same way.
But now, instead of calculating this area, right here, we need
to calculate the area of this wedge, of this little wedge,
of this little sector.
And, this actually is not so difficult to do, because see
the point is that we know the area of the entire disc.
So, if you have a disc of radius r, we know
the entire area.

What's the area of the disc?
Pi times r squared, right.
Area is pi times r squared.
And, okay, so now suppose we are given say 1/4 of the disc.
What is the area of this 1/4 of the disc.
Well, it will be 1/4 of the disc, right.
So that will be 1/4.
Why 1/4, because the angle enclosed here is pi over 2.
So, 1/4 is pi over 2 divided by 2 pi, which is like the
full angle of the disc.
So, in other words, if you want to calculate the area not of
the entire disc, but of part of the disc which is enclosed
within the angle of pi over 2, you have to multiply this
result pi r squared by the ratio of the angle which is
enclosed, to the total area of the disc.
So, it would be pi over 2 divided by 2 pi.
So, that would be 1/4 of pi r square, which of course we knew
in the first place, because clearly this is going to be 1/4
because there are four pieces, which are exactly the same.
But let's suppose you are in a pizza restaurant, and you have
a piece like this who 's angle theta, and they didn't cut it
in a very uniform way, so you don't know exactly.
It's not pi over 4, it's not pi over 6 or pi
over 3, whatever.
And, you want to calculate what area, how much pizza you got,
and then compare it to how much your friends got, say.
Maybe they got more, so they should share maybe,
give you back some.
So, what will be the answer.
Well, we should argue in exactly the same way as before.
Before that angle theta was pi over 2, and what we did, we
took the ratio between pi over 2 and 2 pi.
And, now instead we'll have a ratio of theta to 2 pi.
That's what will represent the piece that we got, as sort of
the ratio of that piece to the entire disc, to
the entire pizza.
And, therefore the area would be pi r squared times theta
divided by 2 pi, 2pi.
Right, So, pi cancels out, and you end up with 1/2
r squared times theta.
That's the area of the sector of angle theta, which is
exactly our question.
Except, in asking this question, we called it delta,
theta instead of theta.
So.
This area is 1/2 r squared delta, theta.
And, that's the analog of this formula, y delta x.
So, just like this formula, y delta x gives rise to this is
integral, this formula for the elementary area of the
elementary wedge of angle delta theta will give rise to the
integral from alpha to beta, 1/2 r squared, d theta.
You see.
So exactly the same, just kind of replace delta
theta by d theta.
And, that's how we get that formula, which I wrote down
at the end of last lecture.
So to summarize, the key to calculating the areas like
this, is to break the figure that you got into small pieces,
which will then look like elementary figures, like in
this case, would be an elementary, a rectangle, which
is, where I magnified and blew it up.
And, I show it there.
Calculated its area and then it gives rise to the formula for
the area of a general picture like this.
Whereas, in the polar coordinate system, you do
exactly the same, but you kind of follow, the shape of the
figure is different, because the coordinates are
now different.
So, now we split into little sectors, like this, where
the angle is delta, tetha.
I magnified one of those pieces right there.
I found the area of that little piece by thinking about pizza,
and calculating it, right, and then I put all these little
pieces together, and this gives rise to this integral.
Any questions about this?
So, that's the way to calculate.
That's the way to calculate the area in full accordance, and
there is a similar formula also for the arc length in polar
coordinates, but the formula for the arc length, I don't
want to spend time on it, because we really take the
formula, which we derived in rectangular of coordinates,
about a week ago.
And, we just made the substitution where x is
equal to r cosine theta and y is equal to
r sin theta And, just see what comes out.
What comes out is some formula in terms r and theta, which you
can then use to calculate the arc length.
So, that's the way we do it for the arc length.
Here, it's more conceptual, because we changed the
point of view slightly.
Instead of looking at rectangles, we look
at this sector.
It wouldn't be wise to try to break this into rectangles,
because they don't fit on this picture.
What fits perfectly in this picture are
these little sectors.
So, then we need to calculate the area of the little sector,
which we've done here.
So, this speaking of pizza analogy, this were the
leftovers from last lecture, and now -- it's not what
you say it's how you say it, you know.
It's a delivery [UNINTELLIGIBLE].
So, I'm working on it.
So, now we go, we move on to our next topic.
And, the next topic is vectors and geometry of space.
And, space here means three-dimensional space
which surrounds us.
You see, up to now in the last three lectures, we talked
about objects which are defined on the plane.
Here, I've drawing curves on the plane.
And, we've discussed various things about those curves, like
arc length, or there is areas which are enclosed
by those curves.
But, the basic point, the most important aspect of what we've
done up to now in this class, is the fact that we have worked
exclusively with objects confined to a plane.
And, by a plane I mean the blackboard, or sort of, an
infinite extension of this blackboard in all directions.
All of our objects have been defined on this plane.
But, now we want to move on, as I told you in the very
beginning where in this class we are going to study not just
objects on the plane, but also objects in space, in
three-dimensional space.
Because, we live in a three-dimensional space.
If you don't count the time.
In fact, we live in four-dimensional space time,
but that will be a topic for the next calculus
class, I guess.
Four-dimensional space.
Now, we'll do, in this class we two-dimensional
space, the plane and three-dimensional space.
So, three-dimensional space is more interesting, because
it fits more objects.
Now it fits not just curves, and there are a lot of
interesting curves, which are, appear in space, which are not
confined in any particular plane, but also perhaps more
interesting, it contains objects of it dimension higher,
namely two-dimensional objects, the surfaces.
Like a sphere for example, or this wastebasket.
These are all surfaces.
If you ignore the fact that they actually have
certain thickness.
So, what we would like to do ultimately in this class is to
understand geometry of those surfaces, as well as curves,
and to be able to answer various questions about them.
Like, finding their areas, and even more sophisticated things.
So, what we need to do now is to sort of lay the
groundwork for this.
In other words, we have to develop some formulas, because
how are we going to represent curves, how are we going to
represent surfaces in a three-dimensional space.
What's the most efficient? [INAUDIBLE]
So, always check for batteries before you begin.
So, in space we need to, you need to develop some
technique in order to attack this problem.
And, the very first thing that we should talk about
is the coordinate system.
Because, you know, on the plane it sort of goes without saying,
that we started out with -- whenever I start writing,
drawing a diagram, drawing a picture, or drawing a curve, I
draw this coordinate system.
kind It's almost like to reflex.
And, usually we call them x and y, the two coordinates.
After last week, we kind of, we're wiser now we realize that
this is actually not the only coordinate system that we have
available for the plane.
For instance, the polar coordinates provides a
different coordinate system.
But this is the basic one, and we should start thinking about
a similar coordinate system in a three-dimensional space.
So, this is the first step that we have to make
on coordinate systems.
Now, of course the problem is that even though I'm
going to talk about the three-dimensional space, I'm
still going to draw on the two-dimensional blackboard.
So, of course, I cannot draw this wastebasket
on this blackboard.
What I can draw is really a projection.
So, I kind of, we kind of create the illusion that
there's some depth in the picture, and we try to imagine
that the three-dimensio, the third dimension is
there somewhere.
So, the way we usually would draw coordinates
would be like this.
So, that, of course the point is that they are not on
the same blackboard.
Only this one and this one lie on this blackboard.
And, this one is sticking out.
It's sticking out, but we're looking at it slightly from
above, and that's why it appears to us this way.
So, now we want to label these coordinates in a certain way.
In the, on the two-dimensional plane, we label them x and y.
And, in fact, some of you may have wondered why we labeled
them x and y and not x and x.
Not in the opposite the opposite.

Why not like this.
Of course, in a sense, the notation is arbitrary.
We have chosen these letters once and for all, so in a way
you can say that it's not a meaningful question,
because it depends on what we mean by x and y.
But, of course what is important here is not so much
which letters we use, but the ordering, the implicit ordering
between x and y; x comes first and y comes second, because
that's how they are ordered in the alphabet, right.
So, we could use y and z, and again it would be assumed that
y goes first and z goes second.
So, there is actually a basic asymmetry between these
two coordinates, x and y.
And, we don't talk about it much, we haven't talked
about this much when we talked about curves.
But, in fact now is a good time to discuss this and to realize
that there's a basic asymmetry.
The point is you could work with a coordinate system like
this, xy, or you could work with a coordinate system yx.
And, these are not equivalent to each other.
There is no way to transform one thing to the other without
removing them from this plane.
Of course, if I were able to -- maybe I should draw it more
like this -- if I were able to picket from the plane and flip
x and y, I would of course then, I will get back to this
picture x and y, that is a more traditional picture
we draw, right.
But, we cannot flip them within the plane.
You can argue, well what if we just try to move this
coordinate like the, you know, on the clock and kind of move
them and make them go through one another.
But, that's not allowed, because they would have to
pass through each other.
And, at that point, they will cease to be a
coordinate system.
They will become parallel to each other.
So that's not allowed.
So, in fact, if you think about it, you will see that there's
no way to transform one into the other.
There's no way to transform one into the other by moves
confined to this plane.
And, that's a very important point, which tells us that
actually our plane has two different orientations.
Choosing one or the other, this coordinate, this coordinate
system or the other coordinate system gives it orientation.
And, the funny thing is suppose that there was another class
behind that blackboard.
And, there were people there like a mirror image of this
one, the bizarre world if you will, and suppose their were
kids , you know, sitting there, and also learning
multi-variable calculus.
Then, this one, one of them, the first one would appear to
them as the second, and conversely.
I mean, not exactly, but if you rotate, that's
exactly what you'd get.
So, in other words the orientation from this side,
there is orientation from this side, when you look at this
blackboard from this side, and there is orientation of this
blackboard from the other side, from the bizarre world side,
and these are two different orientations, there's a two
different worlds, and you can't transform one world into
another within the plane, within the confines
of the plane.
If you go into this three-dimensional space
you can do that.
You can just pick it up, and flip them.
But inside the two-dimensional space, you cannot do that.
At first, it appears that it's a kind of a nuisance, and you
may think that's not a very important point, but actually,
in fact for the plane it's not so important.
But, in three-dimensional space, it becomes
more important.
And, that's why we emphasize it.
In the three-dimensional space, we also have different choices
of writing coordinates, of labeling coordinates, and in
three-dimensions we will label coordinates as xyz.

And, of course again the most important point is not so much
which letters we use, but the ordering in which they appear.
Northern And the ordering, of course, will be the natural
one, 1, 2, 3, just the way they appear in the alphabet.
But, now the question is how do you assign.
There are now many choices to assign these letters
to this coordinate axis.
And, the question is are we going to get always the
same coordinate system.
In other words, are we going to get coordinate systems which
can be transformed one into another, by moves within the
three-dimensiona space.
And, so now since I've shown you that in two-dimensional
space on the plane, that's not true.
In other words, on the plane there are two different
coordinate systems, which cannot be transformed into one
another, you would not be surprised to know that actually
-- here as well -- in three-dimensional space, there
are also inequivalent coordinate systems.
But, you might be surprised to know, that again, that
there are only two choices.
Not more than two.
So the choices are, basically, the one which we will always
use will be like this, xyz.

And, the other choice, which we will not use, which is not
equivalent to this one, which is a bizarre world coordinate
sytem, is this one.
In other words, we switch x and y, just as we did
before on the plane.
Just the way we did before on the plane.
If you tried the other choices, for example, you put xyz like
this, you will see that there are many different ways.
There are six different ways to label them.
You will see that that picture can be transformed by simple
rotation into one of these two pictures.
So, really there are only two equivalent ones up
to rotation within the three-dimensional space.
If we could go into the four-dimensional space.
If there were a fourth dimension that we could just
visualize, just the way we can visualize the third-dimension
when talking about the plane, we would actually be able to
pick it up and transform one coordinate system into another
in that four-dimensional space.
But, since we don't have a four-dimension, or more
precisely the fourth-dimension, time is kind of elusive, it's
not easy to visualize it.
So, within the three-dimensional space
we could not do that.
So, we have to agree from the beginning on what is the rule.
What is the rule for labeling these coordinates, because we
have to do it now, because from now on, we have to
use the same system.
So, that we're on the same page or on the same blackboard.
Now, what is the rule.
And, so the rule is that different ways
to phrase the rule.
And the rule is in the book.
It's explained by using fingers.
But, I always forgeth that.
So, I will tell you the rule which
I use.
Which is, you know, I grew up in Russia, and we use the rule
which is called the Cork Screw Rule.
And this is not to say that Russians like to drink,
[LAUGHTER], but anyways it's called the Cork Screw Rule.
And, the way it works is like this.
If you wrote the cork screw from x to y, so that
here is the cork screw.
I kind of draw the most basic one.
If you rotate it from x to y, then the thing will go in the
direction of the z axis.
In other words if you -- well it's not a good picture
because then the bottle would have to be upside down.
But, you see what I mean.
So, if you think about the cork screw going from x to y, which
is natural right, going from x to y is like going from the
first coordinate to the second.
Going from x, making a rotation as if were moving the x
coordinate system to the y, sorry x-axis to the y-axis,
then the screw will move in this direction of the z-axis.
So, that's the rule which I find easy to remember, even if
I don't have a cork screw on me, it's easy to remember.
But, you can use whatever rule you want, you just have to
remember that there is such a rule, and to draw this
picture in this way.
OK.
So, that's the first point that we have to make about this.
The second point is we have two develop, we have to develop
some tools for representing objects in space.
So, the first, I already told you that the objects will be
mostly interested in -- I'm still looking for a battery,
but I guess no luck.
So, the objects that we are most interested in are
curves and surfaces.
But, in fact there, are simpler more elementary
objects, namely points.
So, before we talk about, you know, these complicated matters
like curves and surfaces, let's talk about points.
In the case of a plane, we know how to represent a point.
We just draw up the perpendicular lines onto
the x- and y-axis, right.
And, say we get x0 and y0, and we represent the points
as x0, y0, like this.
And, we are going to do the same in the
three-dimensional space.
So, I draw the same coordinate system.
And, I'm thinking that OK, if I go from x to y, it's
going through z, so that's how I should put them.
Because if I put them in the other way, I would have to
move the cork screw this way, and it will go down.
So, z would have to go this way.
So, this is a good orientation.
This is orientation we've chosen.
And, our mirror image will have that opposite orientation.
So, if we have a point, we also can represent it by its
coordinates, and now, they're going to be three coordinates,
obviously, because we are in a three-dimensional space.
And, the way we find these coordinates is as follows.
We draw up the perpendicular line from this point
to the plane x and y.
Now, you have to, at this point, you have to use your
imagination, and you really have to think of x and
y as a plane which is sticking out, like this.
And, the point is actually not here, it's not in the
zy plane, but it's here.
So, you've got -- I can't show you because one of my hands is
busy with the microphone, but you can see, I'm trying.
So, this a plane, and the point is here, so you draw up the
perpendicular down, like this.
So, you get a point on the plane, which is the xy plane.
Now, I draw it like this.
So, the point is somewhere in space, and that's
where it drops.
So, now I get a point in the xy plane.
But, a point in the xy plane we already know how to represent.
We just drop the perpendicular lines to the x-axis
and the y-axis.
So, what we got already is y0 and x0 coordinates of
this point, two of the three coordinates.
What about the third coordinate.
It is tempting to draw a perpendicular like this, but
that's not true, right.
To give it really a depth, to give it the illusion of a
three-dimensional picture, I have to do it in
the following way.
I have to connect the origin to this point, which is the point
I got by dropping my original point on the xy plane, and
I have to connect to the z-axis by a parallel line.
So, this one and this one a parallel to each other.
And, that's going to be the z0 coordinate.
You see, the point is that this is perpendicular to this entire
plane, so in particular it's perpendicular to this.
And, that's why the z-axis is perpendicular to this line.
So, what I'm doing really, is that I'm dropping
perpendiculars to each of the three coordinate systems.
But, the way I draw it is slightly complicated, because
I'm using two-dimensional projection of the picture.
It would have been much easier if I had a three-dimensional
model of this whole thing.
So, then you would see more clearly what the coordinates
are, but I think it's fairly self-explanatory here.
If we have this picture, then we would say say that the point
has coordinates x0, y0 and z0.

And, this is incomplete analogy, where the situation
that we have, the picture that we have, in two-dimensions, so
nothing is surprising here.
What else do we need to know, we need to know the distance
between two points, right.
So, for instance, we would like to find oftentimes -- met
me use a different chalk.

Oftentimes, we might want to find the distance from
this point to the origin.
Let's call this R.
So, we need a formula for this distance.
And in two dimensions, we know that this distance is computed
by a very simple formula, namely square root xz, x
squared plus y squared.
And, now the formula is going to be very similar.
It's going to be the squared root of x squared plus y
squared plus z squared, OK.
So, very similar.
Instead of sum of two squares, like this, you have the sum of
three squares, and then you take the squared root.
And that's how you calculate the distance from this
point to this point.
And, it's very easy to prove this by using Pythagoras
Theoreom, and it's in the book, I'm nog going to get into that.
It's very easy.
All right, what if we were to calculate the distance between
two different points.
So, you could have a point b and then you have another point
b prime, and say here you have coordinates x0, y0 and z0, and
here you have coordinates x1, y1 and z1.

So, the distance, so you see the point is that this, find
this distance you can just parallel transport this
interval to the, in such a way that the first point would
end up at the origin.
And, if you do that, the second point will have coordinates
which are just the differences between the coordinates of
the original two points.
So, this point would have coordinates x1 minus x0, y1
minus y0, and z1 minus z0, OK.
So, the distance, let's call it d here.
The distance would be then, according to this formula, the
square root of x1 minus x0 squared plus y1 minus y0
squared plus z1 minus z0 squared, just like this.
Is this clear to everybody?
OK.
And, this already gives us a way to describe a very
important surface in space, namely the sphere.
Because, the sphere is defined as a set of all points, which
are equidistant from a given point.
So, for example, let me erase something.
Let's talk about the sphere or radius r with the
center at the origin.
These are all points, whose distance -- of points P --
whose distance to the origin.
Origin I denote by this letter zero.
This is not to say, I don't mean the number zero, but I
mean this point, it's really not zero, but it's point O.
So, point whose distance to the point O, to the origin, is
equal to R, and R is some fixed number, right.
So, what does it mean.
It means that the squared root of x squared plus y squared
plus z squared, where xy and z are coordinates of the point P
will have to be equal to R.
Because, that's exactly the distance, and if we say that
the distance is equal to R, that the equation we get.
Now, it's a good idea to square both parts, both sides,
left-hand side and the right-hand side.
So, then we get the equation, x squared plus y squared plus z
squared is equal to R squared.
And, that's remarkably similar to the equation of the circle.
So, compare with the circle, circle, for which the
equation is x squared plus y squared equals R squared.
Here, now, we have a third variable.
So, we just add the third square to the left-hand side,
but otherwise the equation looks very similar.
So, now we have the equation of the sphere with,
centered at the origin.
And again, you see, remember the rule which I explained to
you at the very beginning, that if want to calculate the
dimension of an object, you have to take the dimension of
the ambient space, in this case 3, and subtract the
number of equations.
Here, I've written one equation.
So, the dimension drops by 1, and we get a
two-dimensional object.
So, we get a sphere which is two-dimensional,
two-dimensional object.
It is a surface.
Surface, by definition is a two-dimensional object.
And, likewise we can write the equation for a sphere which is
centered not at the origin, but at any given point, by
using this formula.
So, if I say that this is equal to R, that would be sphere of
radius R centered at that point P.
That would be the equation for x1, y1 and z1 with x0, y0
and z0 fixed being the coordinates of the point P.
All right.
So, we already have some rudimentary knowledge of the
three-dimension space, and we know an equation of a
one basic surface in it.
So, what's the next step.
The next step is to introduce objects, to introduce
a new tool.
To introduce objects of a different kind, which you have
not used yet in this course.
These objects are called vectors.
And, you will see that these objects are very useful for,
precisely for studying the kind of questions that
will be interested in.
Like, finding areas representing
objects, and so on.
So, the next step is to talk about vectors.

So, how many of you know vectors?
Oh, all right, so maybe we should just go home.
OK.
But, let's just go briefly, let me give you an overview.
So, I will not dwell on it too much then.
So, what is a vector.
A vector is an object which is the which is different
from numbers, which we have studied so far.
So far, we studied different objects, we studied, we know
numbers, we know points, for example, on the plane.
Points on the plane are numbers, they are represented
not by single numbers, but by two numbers, right.
And, in fact they are, they're more somehow two numbers,
because a point is a geometric object which is positioned
somewhere on the plane or in space.
So, a point is something which has a position.
Which is a position on the plane or in space.
Ad, a vector is an object of yet another kind, which is
characterized by two properties.
It has a magnitude or length, and it also has a direction.
So, we usually draw vectors as directed line segments.
So, it is convenient to draw a vector as a line segment
like this, connecting A point a to point B.
And, you see when I do that, I give it two properties.
One is a direction, because this arrow points
in that direction.
And, the second is a magnitude for the length.
And , the magnitude is the distance between the points,
point A and point B.
Now, the question is how do we, how are we going to represent
vectors in practice.
In other words, in practice we will represent points by their
coordinates, so we should discuss also how to present
vectors by something like coordinates, and various
separation on vectors.
What kind of things can we do with vectors.
And, here there is a subtle point, which is very important.
Which is that in fact, a vector is not really a picture like
this but, vector is really what I said, something which is
determined by its direction and its magnitude.
In other words, if you take a parallel transport of this.
If you just move it to another pointed, directed segment like
this, we say points A prime and b prime.
And, you make sure that the directions are
exactly the same.
In other words, if these two lines are parallel and they
point in the same direction, and the lengths are the same.
And, actually we will not distinguish between these
two vectors, you see.
We will not distinguish between them.
So, a vector is not really a pointed, pointed segment
or directed segment.
It's a whole class of directed segments.
A vector is, this is one representative of this vector.
This is one possible way to represent our vector, as a
directed segment going from the point A.
But, you might as well start at the point A prime and you would
represent exactly the same vector.
So, usually we will denote vector by small Latin letter
like a, and we'll put an arrow above it.
And, in the first approximation, you can really
think of a vector as just an arrow like this, which has
direction in magnitude.
But, you have to realize that many different arrows represent
the same vector, namely all parallel arrows, parallel to
each other, pointing in the same direction, of the same
length, represents the same vector.
So, then of course the question is how can we possibly work
with these guys, because there are so many possible
ways to represent them.
Well, that's where we have to start using
our coordinate system.

So, we draw our coordinate system.
And, again remember the xyz.
Now, as I said the same vector A could be represented
in many different ways.
But, find such a way, we have to pick the initial point.
One we pick the initial point then it's already, the arrow
is determined by that.
If we're drawing that vector A.
But, amongst all the points on the plane, once we introduce
the coordinate system, there is the origin, there
is a special point.
So, we might as well draw a representative from this class,
from this whole set of possible arrows, representing
this given vector.
We can draw one of them, which starts at the point, at
the origin of this three-dimensional
coordinate system.
So, the result is, and I I'm trying to draw it parallel
to this one, in some sense.
But, although in my previous discussion I kind of, I was
kind of working on the plane, but now I'm working in space.
So, in fact this endpoint is not necessarily on the plane,
it's not necessarily part of the blackboard, but it
cold be, again, hanging somewhere in the middle.
Like on this picture.
So, now the vector A gets a much more concrete realization.
It is a pointed interval or directed interval,
or directed segment.
But, now it goes from the point O from the origin
to some point P.
So, that's already much less ambiguous.
In other words, once you introduce the coordinate
system, you have a preferred representative for all
possible arrows, giving you the same vector.
And, then what we'll do is we will write that
a is equal to OP.
Or, in this case we would write that A is equal to
AB or A prime, B prime.
In other words, different notation in which you use the
initial point and the endpoint and again put the arrow on top.
So, from this point of view, it looks like a vector is
essentially just determined by its endpoint.
Because, we agree to use as the initial point, the origin.
So, the only freedom we have is where b to put the endpoint.
So it looks, at first glance, it looks like there's no
difference between a vector and a point.
But, this is misleading.
You have to realize that this is misleading, and actually
there is a lot more to the vector than there
is to a point.
Even though, when we represent it in this way, the vector will
essentially be determined by its endpoint once you center it
or make the initial point to be at the origin.
And, the main difference is in the meaning of this, in the
physical meaning, physical interpretation of a vector,
as oppose to a point.
A point is just a point.
It's located somewhere, and it knows nothing about
the rest of space.
But, vector really is something else.
Here, we talk about a vector being a pointed interval.
But, in fact, a much better way to think about it, of
a vector, it's much better to think that vector is a
transformation of space.
A vector is a shift of space.
Beause, what do you need to know if you want to shift
something. what do you need to know.
Well, it will be much easier to express this of a instead
three-dimensional space, it will be much easier to
explain this on the plane.
So, let's use this analogy.
So, I have this table, I have this chair, OK.
So, it's on the plane.
So think of it as a kind of representation of a point on
the plane, where plane is the floor, or the
podium really, OK.
So, I want to shift it, so I move it in this way.
What do I need to know, what do I need to say in order
to explain how to move it.
In other words, if I ask one of you to move it, I
have to explain in which direction and how far.
So, and suppose I ask you to shift everything.
In other words, I can ask some workers to come here and shift
this podium. [UNINTELLIGIBLE]
Say, I want to be closer to you, so I want to move this
podium, whatever, 1 yard closer.

What does it mean.
It means that each point of this podium will be moved by 1
yard in the same direction.
That's the vector.
A vector is a rule by which you shift all points.
And, then if you look at a particular point, like a point
A, then a vector will displace it or shift it to a appoint B.
If you look at another point, A prime, this same vector, this
same rule will shift it to a point B prime.
And, the origin will shift it to this point P.
So, this is a much better way to think of a vector as a rule
for displacement, for sort of parallel transport, or for
the entire plane, or the entire space.
If you have such a rule, then you know where each point
goes, and that's what each arrow represents.
And, now it becomes much more clear why different arrows, as
long as they are parallel and have the same magnitude
correspond to the same vector, because they are part of
the same rule, part of the same shift rule.
You see what I mean?
Any questions about this?
This is an important point to remember.
To kind of understand better what's, the kind of stuff that
we're going to do with vectors.
All right.
That's sort of physical interpretation.
But now, more concretely, we would like to work with
vectors, and try to represent them in much the same way as
we represent points, say.
So, a point has three coordinates.
So, now we want to represent vectors also in a very
concrete, algebraic way.
What I talked about up to now is kind of geometry, I drew
pictures and I explained the geometric meaning, but now I
would like to do some algebra, I want to represent my vectors
by some coordinates, or some algebraic objects.
And, it's clear what we need to do.
We simply need to place the vector, the initial point of
the vector at the origin, and we have to keep track
of the endpoint.
And the endpoint, we already learned how to describe.
We describe it by its three coordinates,
x0, y0 and z0, right.
So, if that's the case, we will write the following.
We will write that A is equal to x0, y0 and z0.

So, what happened.
We're using exactly the same information as the information
provided by the to point P.
But, because now the point P is really not the central object,
but it's something which is just, which is just sort of an
auxiliary object, which we just used to understand the vector,
because that's theendpoint for the pointed interval which we
obtained when we applied our displacement at the
origin, right.
So, the point P sort of plays a secondary role here.
Nevertheless, the point P has a very nice representation
by x0, y0 and z0.
So, we might as well use this information, because it
uniquely determines our vector.
Because, we agreed that the initial point is at the origin.
So, all we need to know it the endpoint.
And, the endpoint is just this.
So, that's why this information is sufficient to
represent a given vector.
But, when the write it like this, we want to distinguish
this notation from the notation which I used
for the point P itself.
For the point P itself, we used notation with brown brackets.
We don't want to write it here like this, because if we were
to write like this, we would be saying that this is a point,
but it's not a point, as I keep trying to explain to you.
It's not really a point.
The point in question here, namely the point P, is really
just the endpoint of the vecor.
But, the vector has a lot more, carries a lot more information.
or has a different geometric interpretation than
just the point.
So, we have to separate the notation for vector, and the
notation for the point.
And, the way we do it, we use this angular brackets instead
of a set of the round brackets.
And to someone, it may appear like we are too pedantic,
and it may appear like what's the difference.
Well the difference is, is we will see now, that
vectors actually have a lot of structure.
We can add two vectors to each other.
We can multiply a vector by a scaler and so on.
A point, a point, doesn't have such a structure, or points
don't have such structures.
We're not allowed to add two points.
We're not allowed to multiply a point by a number.
A point is just a point.
It's static, it just sits there.
A vector is kind of a dynamic thing, which like I said, is a
transformation of a space, of shifting everything the same
direction and by the same magnitude.
So, if you have two vectors, two such shifts, you can
actually do them, you know, one after another, and therefore
you get a third vector, you get the sum of the two
vectors, and so on.
So, in that sense a vector really is a different
object on the point.
And, that's why we really have to distinguish between vectors
and points, and use a different notation.
So this is notation number one.
Notation number two.
We, amongst all vectors that we have on the plane, we
introduced the basic ones.
The basic ones are the vectors which go from the origin in
the direction of one of the three axis.
I mean, from the origin, in case you want to
apply to the origin.
But, in fact, as I said, you can apply it to any
other point as well.
So, these are the following three vectors.
One of them goes along this axis, and has length
one, and it's uniquely determined by this.
Remember, vector is uniquely determined by direction
and the magnitude.
So, amongst all the directions in space, once we introduce
coordinate system, you have three basic
directions, xy and z.
So, why not use those three vectors.
But, then you have to say which magnitude, the simplest
magnitude would be one.
So, you got yourself three vectors: this one, this
one, and this one.
And, we'll call them i, j and k, OK.
And, so the point is that we can also write this vector as
x0 times i plus y0 times j plus z0 times k.
In other words, we can decompose any vector as a
combination of multiples of the three the basic vectors.
Which of course, immediately begs the question as to what do
I mean by addition, by the addition of vectors, and what
do I mean by multiplication of a vector by a scaler.
But, I'm kind of slightly jumping ahead, because I
know that you already know most of this material.
Let me just remind you of a simple rule of how
to add two vectors.
There is a so-called parallelogram rule,
do you remember the parallelogram rule?
Yes or no?
No?
Someone said yes, but most people don't remember?
Should I remind you?
I remember. [LAUGHTER].
OK.
So, the rule is like this, if you have two vectors, A and B,
and you want to calculate what is A plus B.
And, the way you do it is you apply the two vectors
to the same point.
Again, vector is a displacement rule.
So, each point knows where it goes, right.
So, now we have two vectors, so we have two rules.
So, let's apply both of them to the same point.
And, it depends in the problem you're solving there will be
some natural point to which you would want to apply.
Now, to find A plus B what you need to is you need to draw
a parallelogram, which is spanned by these two vectors.
In other words, you draw a parallel line to this one,
parallel to this vector.
And, at this point you draw a parallel line to this vector.
So, this and this are parallel, and this and this are parallel.
And, then these two lines will intersect somewhere.
And, that's your, that's the endpoint of the sum.
So, that's the vector A plus B.
If this is A and this is B, then that's A plus B.
Explanation of this rule.
Like I said, it's better to think of a vector as
a displacement rule.
So, now you've got yourself two displacement rules, A and B.
Each of these rules can be applied to any
point whatsoever.
OK.
Let's apply, first let's apply first of them A and then
apply B, consecutively.
Let's apply them one after the other.
So, first we apply A.
Now, we have to say to which point do we apply it.
Well, let's apply it this point.
The end result is that we get here, to this point.
After this, I would like to apply the displacement rule B.
And, like is I said the displacement rule B can
be applied at any point.
Here, I drew the result of its application to this point.
What will be the result of its application to this point.
Well, I have to take a parallel line of the same length.
This will be a directed interval, which will represent
the same vector, right.
So, I end up, the net result of these two displacements,
this one and this one, I end up at this point.
That's why the composition of the two or the sum of the
two vectors is this vector.
So, it's very simple geometric interpretation.
If you draw the picture using this parallelogram, this is
alled parallelogram rule.
If you draw it using this triangle, it's called
the triangle rule.
But, they the essence, the meaning is the same.
Is this clear?
OK.
We can also multiply vectors by scalers.
When I say scaler, I just mean usual numbers, real numbers.
What does it mean.
Well, here we should think, we should think of a vector as
a direction and magnitude.
When we multiply by scaler, we, well, it depends on
what kind of scaler.
Let me rephrase it.
If you multiply by a positive number, if you multiply by a
positive number, this will be a vector in the same direction.
And, the magnitude will be c times the magnitude
of the original vector.
So, the picture will be something like this.
This is the original vector.
Let's say you want to take, this is A, and you want 2A,
2A is going to be like this.
So, this distance will be twice the distance of A, right, and
you have the same direction.
If you want negative two, or if you want any negative number,
it will have to be the opposite direction.
So, this for example is negative 2A.
So, same direction,
c less than 0, opposite direction.
Opposite direction.

So, you have two operations.
Addition of two vectors and multiplication
of a vector by a
scaler.
And, here I combined this two operations.
I take my vector i, and multiply it by x0.
What do I get.
Well, according to the rule, I have to get something which
goes again along the x-axis, and now has length which is
equal to x0 times the lengths of this guy.
But, the lengths of this guy, just like the lengths of the
second guy and the third guy j and k, is equal to 1.
So, when I multiply by x0, I get something
which has length x0.
So, it's just the vector which ends at this point.
And, likewise if I take j time y0, I get the vector
which ends at this point.
And, if I take the third one k times z0, I get this one.
And, now, it's not difficult to see that if you take sum of
this vector, this vector, this vector, you will get this one.
I'll leave it for you to figure this out.
So, that's how you get the second, the second
representation of a vector.
This is the first one, where we just keep track of the
coordinates of the endpoint.
This is a little bit more descriptive, because it really
emphasises the fact that it is the displacement is a
superposition of three displacements: one in the x
direction, one in the y directionand one in
the z direction.
All right, so what's next.
Next we have to work out some tools for dealing with vectors,
and these tools are the so-called dot product
and cross product.
So how many people know what dot product is?
So, I'll go quickly over this.
By the way, an important point is that I try to do all of
this in full generality.
In other words, in a three-dimensional space.
So, you see that I'm writing everything in space.
I have three coordinates, my vector has three
components and so.
But, you can do the same analysis on the plane.
If you are on the plane, if you are on the plane, you can also
represent vectors in a very similar way.
The only thing that will be different is that you will be
missing the last coordinate, the last component.
That's all, right.
Because now, you will have two coordinates x and y.
You can still talk about vectors on the plane, and you
can transport this vector in such a way that its initial
point is the origin.
And, then you will have two coordinates, you have some
point P, you'll have two coordinates x0, y0, so your
vector, let's call it again A, this vector A will
be written as x0, y0.
So, you see, just two components instead of three.
So, if you do a homework exercise, and you are given a
vector in this form, that's how you know right away, that it is
a vector on the plane and not in space.
Because, you only have two components.
Or, you can also write it as i, x0 times i plus y0 times j,
where one more time, i and j are unit vectors in the x
direction and the y direction.
So same kind of direction on the plane as in space.
Now, I will talk about dot product and cross product.
And, dot product also is preferably well
defined on the plane.
Formulas are very similar, but I will not write then, they
will be easy to derive by using the formulas in the
three-dimensional space.
The cross product, the next operation that we'll
introduce only make sense in space not on the plane.
So, we'll only work with it in space.
OK.
So what's the dot product.
First, comes the dot product.
So, you've got two vectors, and the dot product is a certain
operation, OK, it's a certain operation, where you're given
two vectors, and you produce a number.
So, this is very important.
You start with two vectors, but you get not a vector,
but you get a number.
This is different from say taking the sum of two vectors.
The sum of two vectors is again a vector.
You start with two vectors, you get a vector.
Also, a scale or multiple, or a multiplication by
a number, c times a.
You start with a number and a vector, and the
end result is a vector.
So, when you think about all these operations -- additions,
scale or multiplication, dot product, cross product --
the first thing you should remember is what is the input
and what is the output.
Usually has two inputs, and one output, like sum.
In addition, two input, both are vectors, output
also vector, right.
So, dot product, likewise is an operation.
You know, you can think of it as a kind of black-box.
It's a black-box, because there are two input and
there is one output.
So, there is some rule, something happens here.
So it eats two vectors, and spits out a number.
OK.
So, the two inputs are vector and another vector, vector
1 and vector 2, and output is a number.
So, I have to give you the rule of how this black-box works.
And here, the important point is that there are two ways
to represent this rule.
Which is actually very nice, because then you can compare
them, and you can derive some useful information about
vectors, which we'll use.
So, what is the rule for the dot product.
So the rule, rule number one, is I would like to draw them
on the plane or in space.
So, I always draw everything on the plane, because that's all
I got, I got the blackboard.
And, thank goodness, because if I had a three-dimensional
thing, it would be much more difficult.
Now, I have this two vectors, a and b, and I would like to give
you a rule how to produce a number out of them.
And, so there are two rules, and rule number one we'll only
use geometric information about them, so it will be kind
of geometric formula.
So, what I'll use is the magnitude of the two vectors,
you see this one has a magnitude, which again is just
the length of this pointed interval.
And, this one also has a length or magnitude.
Sometimes I call it the length some times I call it the
magnitude, it's the same thing.
And, there is one more piece of geometric information, namely
the angle between them.
Now, it's a sharp angle, but actually in general it could be
obtuse angle, they could even be opposite to each other, but
anyway, there is an angle between them.
So, the dot product, first of all the notation for it is if
this one is a and this one is b, this is called a dot
b, which is why it's called a dot product.
All right.
So, what is it.
It is the length of the first one, times the length of the
second one, times the cosine of the angle between them.
This is a vector This is a vector.
This is a number, this is a number, this is a number.
I take the product of these three numbers, so this
is a number as promised.
I start with two vectors and I get a number.
That's the dot product.
Rule number 2.
Up to now, we have not used algebraic representation
of a vector.
An algebraic representation of a vector has two-- you know, we
have two different ways to package this information about
the vector by using this three numbers: x0, y0 and z0, which
is obtained in this way.
So, let's suppose that a is x1, y1 and z1.

Let me write it on a different board, so I don't have to.

Then rule number 2, I might as well write it there.
Rule number 2 that a dot b is x1 times x2 plus y1 times y2
plus z1 times z2 In other words, I take x-coordinates
and multiply them.
I take the y-coordinates and multiply them, and I take the
z-coordinates and multiply them.
It's very easy, just like this, multiply, multiply,
multiply, add them up.
That's the number I get.
And, it looks sort of a miracle the fact that the two
definitions, two rules, are actually equivalent
to each other.
And, But, it's actually very easy to prove.
It's very easy to prove that this implies this by
using this presentation.
Actually, it might be worth wild doing this.
But, actually the funny thing is in the book, they explain it
in the opposite direction, From rule 2 to rule 1, which
is much more difficult.
So, I think it's actually much better to take the first
rule as a definition.
And, that's often the case.
The best definition is geometric, because it has a
nice interpretation, and usually has a much deeper
meaning, like here.
It's something which has to do with the structure of the
vectors and the position of the vectors by the way they
are situated on the plane or in space.
Whereas, the second definition is algebraic.
And, it's really a working definition of something which
is very useful in calculation, but might not be so useful for
conceptual understanding of what we are calculating.
So, usually the right way to go is from the geometric
definition to the algebraic definition.
And, that's the way it works here.
So, you see, and I would like, I think it's a good idea to
actually to do this, to explain this to you, because here I
will illustrate different rules of dot product in regards to
addition and multiplication of vectors.
So, you see here is what I'm going to do.
I'm going to rewrite this in a second way.
I'm going to write it as x1, i plus y1, j plus z1, k, and
likewise for the second one.

So far, I've done nothing.
I'm just using an equivalent notation.
But, now you will see how this is actually better, because
what I'm going to do, is I'm going to calculate the dot
product by using the second definition.

OK.
So, now it looks like a product of two things, and it's
tempting to open the brackets.
The way we would normally open the brackets if we were
calculating with numbers.
Now, [UNINTELLIGIBLE]
is not clear that this is allowed.
But, in fact, you can justify this by using rule 1.
So, the point is that actually we are allowed to open the
brackets in the same way as we open the brackets when we do
calculations with numbers.
So, when we opened the brackets, we're going to end up
with each of these terms, on each of the terms in the first
sum dot one of the terms in the other side, right.
So, this is going to be x1, i dot x2, i.

Now, in principle I have to write down how many,
3 times 3 is 9 terms.
OK.
But, I'm going to save some time.
So, I'm going to see the following, that there will be
terms of two different kinds.
In one of them, they will have matching vectors, like i to i.
There will be three of them x1, i dot x2, i, y1, j dot
y2, j and z1, k dot z2 k.
Right.
So, these are the three summons that I will get, which will
involve the same two vectors, ii, jj, kk, and then I
will have cross terms.
I will have terms which would involve say i and
j, i and k and so on.
So, let me just write the first one of them, x1, i, and you
will see why I will not need to write the rest of them, x1, i
say times y2, j, and so on I'm skipping five more terms, but
all of them are going to be cross terms.

So, now I have to calculate each of this.
I'm going to calculate each of this terms, each of this
nine terms, separately.
And for this, I'm going to use the following rule, if
I have x1, i say dot x2, i.
I can pull out the numbers, I can pull out the scalers out.
I can pull out the scalers and put them outside of
the dot product, right.
So, that's the same as x1, x2 times dot product of i and i.
So, what's the dot product of i and i; i dot i.
Here's my vector i, and I want to take dot
product with itself.
Now, the rule is that this is going to be equal to the
lengths of i times the length of i times the cosine of
the angle between them.
But, the angle between them now is 0.
Right.
Because, I take the same vector twice.
So, this is one, this is one, and the cosine of 0 is 1.
So, this is 1.
The dot product of i with i is 1.
So, the net result is x1, x2.
So, this first term gives me x1, x2.
Likewise, the second term, y1, j dot y2, j will
give me y1, y2.
Because, I will need to calculate the dot
product j dot j.
And, that's again 1.
Because, j is a unit vector.
Same calculation.
And, finally z1,k dot z2, k will be z1, z1.

So, the first three terms will give me exactly the answer,
which appears in rule 2; x1, x2 plus y1, y2 plus z1, z2.

What about the cross terms.
Let me calculate the first one of them; x1, i dot y2, j that's
x1, y2 times the dot product of i and j.
And, what's the dot product of i and j.
That's 0, right.
And, the reason is that now the angle between them is
pi over 2, and the cosine of pi over 2 is 0.
So, when I apply rule 1, I see that i dot j is 0.
So, all the cross terms will disappear, and I will end
up with rule number 2.
So, you see it's very easy to go from rule
1 to rule 2 this way.
And now, the main point is that you can actually use, you can
put these two rules together, and this will allow you to find
the cosine of the angle between two vectors, when you just,
once you know the three components.
Because, you can turn this around, and you can say that
the cosine of the angle can be written as the ratio
of the dot product.
What is this time?
I think I have one more minute, no.
OK.
Well, you got the idea.
We can turn this around and calculate the cosine from
knowing the lengths and dot product.
OK.
We'll continue on Thursday.