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So let's start right away with
stuff that we will need to see
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before we can go on to more
advanced things.
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So, hopefully yesterday in
recitation, you heard a bit
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about vectors.
How many of you actually knew
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about vectors before that?
OK, that's the vast majority.
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If you are not one of those
people, well,
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hopefully you'll learn about
vectors right now.
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I'm sorry that the learning
curve will be a bit steeper for
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the first week.
But hopefully,
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you'll adjust fine.
If you have trouble with
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vectors, do go to your
recitation instructor's office
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hours for extra practice if you
feel the need to.
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You will see it's pretty easy.
So, just to remind you,
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a vector is a quantity that has
both a direction and a magnitude
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of length.
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So -- So, concretely the way
you draw a vector is by some
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arrow, like that,
OK?
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And so, it has a length,
and it's pointing in some
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direction.
And, so, now,
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the way that we compute things
with vectors,
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typically, as we introduce a
coordinate system.
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So, if we are in the plane,
x-y-axis, if we are in space,
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x-y-z axis.
So, usually I will try to draw
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my x-y-z axis consistently to
look like this.
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And then, I can represent my
vector in terms of its
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components along the coordinate
axis.
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So, that means when I have this
row, I can ask,
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how much does it go in the x
direction?
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How much does it go in the y
direction?
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How much does it go in the z
direction?
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And, so, let's call this a
vector A.
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So, it's more convention.
When we have a vector quantity,
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we put an arrow on top to
remind us that it's a vector.
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If it's in the textbook,
then sometimes it's in bold
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because it's easier to typeset.
If you've tried in your
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favorite word processor,
bold is easy and vectors are
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not easy.
So, the vector you can try to
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decompose terms of unit vectors
directed along the coordinate
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axis.
So, the convention is there is
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a vector that we call
***amp***lt;i***amp***gt;
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hat that points along the x
axis and has length one.
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There's a vector called
***amp***lt;j***amp***gt;
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hat that does the same along
the y axis,
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and the
***amp***lt;k***amp***gt;
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hat that does the same along
the z axis.
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And, so, we can express any
vector in terms of its
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components.
So, the other notation is
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***amp***lt;a1,
a2, a3 ***amp***gt;
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between these square brackets.
Well, in angular brackets.
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So, the length of a vector we
denote by, if you want,
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it's the same notation as the
absolute value.
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So, that's going to be a
number, as we say,
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now, a scalar quantity.
OK, so, a scalar quantity is a
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usual numerical quantity as
opposed to a vector quantity.
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And, its direction is sometimes
called dir A,
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and that can be obtained just
by scaling the vector down to
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unit length,
for example,
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by dividing it by its length.
So -- Well, there's a lot of
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notation to be learned.
So, for example,
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if I have two points,
P and Q, then I can draw a
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vector from P to Q.
And, that vector is called
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vector PQ, OK?
So, maybe we'll call it A.
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But, a vector doesn't really
have, necessarily,
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a starting point and an ending
point.
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OK, so if I decide to start
here and I go by the same
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distance in the same direction,
this is also vector A.
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It's the same thing.
So, a lot of vectors we'll draw
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starting at the origin,
but we don't have to.
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So, let's just check and see
how things went in recitation.
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So, let's say that I give you
the vector
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***amp***lt;3,2,1***amp***gt;.
And so, what do you think about
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the length of this vector?
OK, I see an answer forming.
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So, a lot of you are answering
the same thing.
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Maybe it shouldn't spoil it for
those who haven't given it yet.
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OK, I think the overwhelming
vote is in favor of answer
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number two.
I see some sixes, I don't know.
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That's a perfectly good answer,
too, but hopefully in a few
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minutes it won't be I don't know
anymore.
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So, let's see.
How do we find -- -- the length
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of a vector three,
two, one?
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Well, so, this vector,
A, it comes towards us along
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the x axis by three units.
It goes to the right along the
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y axis by two units,
and then it goes up by one unit
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along the z axis.
OK, so, it's pointing towards
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here.
That's pretty hard to draw.
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So, how do we get its length?
Well, maybe we can start with
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something easier,
the length of the vector in the
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plane.
So, observe that A is obtained
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from a vector,
B, in the plane.
00:07:04.000 --> 00:07:09.000
Say, B equals three (i) hat
plus two (j) hat.
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And then, we just have to,
still, go up by one unit,
00:07:15.000 --> 00:07:17.000
OK?
So, let me try to draw a
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picture in this vertical plane
that contains A and B.
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If I draw it in the vertical
plane,
00:07:23.000 --> 00:07:27.000
so, that's the Z axis,
that's not any particular axis,
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then my vector B will go here,
and my vector A will go above
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it.
And here, that's one unit.
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And, here I have a right angle.
So, I can use the Pythagorean
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theorem to find that length A^2
equals length B^2 plus one.
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Now, we are reduced to finding
the length of B.
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The length of B,
we can again find using the
00:08:02.000 --> 00:08:06.000
Pythagorean theorem in the XY
plane because here we have the
00:08:06.000 --> 00:08:09.000
right angle.
Here we have three units,
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and here we have two units.
OK, so, if you do the
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calculations,
you will see that,
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well, length of B is square
root of (3^2 2^2),
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that's 13.
So, the square root of 13 -- --
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and length of A is square root
of length B^2 plus one (square
00:08:32.000 --> 00:08:41.000
it if you want) which is going
to be square root of 13 plus one
00:08:41.000 --> 00:08:49.000
is the square root of 14,
hence, answer number two which
00:08:49.000 --> 00:08:54.000
almost all of you gave.
OK, so the general formula,
00:08:54.000 --> 00:09:02.000
if you follow it with it,
in general if we have a vector
00:09:02.000 --> 00:09:07.000
with components a1,
a2, a3,
00:09:07.000 --> 00:09:16.000
then the length of A is the
square root of a1^2 plus a2^2
00:09:16.000 --> 00:09:23.000
plus a3^2.
OK, any questions about that?
00:09:23.000 --> 00:09:29.000
Yes?
Yes.
00:09:29.000 --> 00:09:32.000
So, in general,
we indeed can consider vectors
00:09:32.000 --> 00:09:36.000
in abstract spaces that have any
number of coordinates.
00:09:36.000 --> 00:09:38.000
And that you have more
components.
00:09:38.000 --> 00:09:40.000
In this class,
we'll mostly see vectors with
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two or three components because
they are easier to draw,
00:09:44.000 --> 00:09:47.000
and because a lot of the math
that we'll see works exactly the
00:09:47.000 --> 00:09:50.000
same way whether you have three
variables or a million
00:09:50.000 --> 00:09:52.000
variables.
If we had a factor with more
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components, then we would have a
lot of trouble drawing it.
00:09:55.000 --> 00:09:58.000
But we could still define its
length in the same way,
00:09:58.000 --> 00:10:01.000
by summing the squares of the
components.
00:10:01.000 --> 00:10:04.000
So, I'm sorry to say that here,
multi-variable,
00:10:04.000 --> 00:10:07.000
multi will mean mostly two or
three.
00:10:07.000 --> 00:10:13.000
But, be assured that it works
just the same way if you have
00:10:13.000 --> 00:10:20.000
10,000 variables.
Just, calculations are longer.
00:10:20.000 --> 00:10:28.000
OK, more questions?
So, what else can we do with
00:10:28.000 --> 00:10:31.000
vectors?
Well, another thing that I'm
00:10:31.000 --> 00:10:35.000
sure you know how to do with
vectors is to add them to scale
00:10:35.000 --> 00:10:39.000
them.
So, vector addition,
00:10:39.000 --> 00:10:48.000
so, if you have two vectors,
A and B, then you can form,
00:10:48.000 --> 00:10:52.000
their sum, A plus B.
How do we do that?
00:10:52.000 --> 00:10:54.000
Well, first,
I should tell you,
00:10:54.000 --> 00:10:56.000
vectors, they have this double
life.
00:10:56.000 --> 00:10:59.000
They are, at the same time,
geometric objects that we can
00:10:59.000 --> 00:11:02.000
draw like this in pictures,
and there are also
00:11:02.000 --> 00:11:06.000
computational objects that we
can represent by numbers.
00:11:06.000 --> 00:11:09.000
So, every question about
vectors will have two answers,
00:11:09.000 --> 00:11:11.000
one geometric,
and one numerical.
00:11:11.000 --> 00:11:14.000
OK, so let's start with the
geometric.
00:11:14.000 --> 00:11:17.000
So, let's say that I have two
vectors, A and B,
00:11:17.000 --> 00:11:21.000
given to me.
And, let's say that I thought
00:11:21.000 --> 00:11:24.000
of drawing them at the same
place to start with.
00:11:24.000 --> 00:11:28.000
Well, to take the sum,
what I should do is actually
00:11:28.000 --> 00:11:33.000
move B so that it starts at the
end of A, at the head of A.
00:11:33.000 --> 00:11:38.000
OK, so this is, again, vector B.
So, observe,
00:11:38.000 --> 00:11:41.000
this actually forms,
now, a parallelogram,
00:11:41.000 --> 00:11:43.000
right?
So, this side is,
00:11:43.000 --> 00:11:48.000
again, vector A.
And now, if we take the
00:11:48.000 --> 00:11:57.000
diagonal of that parallelogram,
this is what we call A plus B,
00:11:57.000 --> 00:12:00.000
OK, so, the idea being that to
move along A plus B,
00:12:00.000 --> 00:12:03.000
it's the same as to move first
along A and then along B,
00:12:03.000 --> 00:12:09.000
or, along B, then along A.
A plus B equals B plus A.
00:12:09.000 --> 00:12:13.000
OK, now, if we do it
numerically,
00:12:13.000 --> 00:12:19.000
then all you do is you just add
the first component of A with
00:12:19.000 --> 00:12:23.000
the first component of B,
the second with the second,
00:12:23.000 --> 00:12:28.000
and the third with the third.
OK, say that A was
00:12:28.000 --> 00:12:31.000
***amp***lt;a1,
a2, a3***amp***gt;
00:12:31.000 --> 00:12:35.000
B was ***amp***lt;b1,
b2, b3***amp***gt;,
00:12:35.000 --> 00:12:40.000
then you just add this way.
OK, so it's pretty
00:12:40.000 --> 00:12:44.000
straightforward.
So, for example,
00:12:44.000 --> 00:12:48.000
I said that my vector over
there, its components are three,
00:12:48.000 --> 00:12:54.000
two, one.
But, I also wrote it as 3i 2j k.
00:12:54.000 --> 00:12:57.000
What does that mean?
OK, so I need to tell you first
00:12:57.000 --> 00:13:06.000
about multiplying by a scalar.
So, this is about addition.
00:13:06.000 --> 00:13:11.000
So, multiplication by a scalar,
it's very easy.
00:13:11.000 --> 00:13:15.000
If you have a vector,
A, then you can form a vector
00:13:15.000 --> 00:13:20.000
2A just by making it go twice as
far in the same direction.
00:13:20.000 --> 00:13:24.000
Or, we can make half A more
modestly.
00:13:24.000 --> 00:13:31.000
We can even make minus A,
and so on.
00:13:31.000 --> 00:13:35.000
So now, you see,
if I do the calculation,
00:13:35.000 --> 00:13:38.000
3i 2j k, well,
what does it mean?
00:13:38.000 --> 00:13:43.000
3i is just going to go along
the x axis, but by distance of
00:13:43.000 --> 00:13:47.000
three instead of one.
And then, 2j goes two units
00:13:47.000 --> 00:13:51.000
along the y axis,
and k goes up by one unit.
00:13:51.000 --> 00:13:54.000
Well, if you add these
together, you will go from the
00:13:54.000 --> 00:13:58.000
origin, then along the x axis,
then parallel to the y axis,
00:13:58.000 --> 00:14:02.000
and then up.
And, you will end up,
00:14:02.000 --> 00:14:05.000
indeed, at the endpoint of a
vector.
00:14:05.000 --> 00:14:19.000
OK, any questions at this point?
Yes?
00:14:19.000 --> 00:14:21.000
Exactly.
To add vectors geometrically,
00:14:21.000 --> 00:14:25.000
you just put the head of the
first vector and the tail of the
00:14:25.000 --> 00:14:30.000
second vector in the same place.
And then, it's head to tail
00:14:30.000 --> 00:14:35.000
addition.
Any other questions?
00:14:35.000 --> 00:14:41.000
Yes?
That's correct.
00:14:41.000 --> 00:14:43.000
If you subtract two vectors,
that just means you add the
00:14:43.000 --> 00:14:45.000
opposite of a vector.
So, for example,
00:14:45.000 --> 00:14:49.000
if I wanted to do A minus B,
I would first go along A and
00:14:49.000 --> 00:14:52.000
then along minus B,
which would take me somewhere
00:14:52.000 --> 00:14:55.000
over there, OK?
So, A minus B,
00:14:55.000 --> 00:15:01.000
if you want,
would go from here to here.
00:15:01.000 --> 00:15:08.000
OK, so hopefully you've kind of
seen that stuff either before in
00:15:08.000 --> 00:15:13.000
your lives, or at least
yesterday.
00:15:13.000 --> 00:15:23.000
So, I'm going to use that as an
excuse to move quickly forward.
00:15:23.000 --> 00:15:28.000
So, now we are going to learn a
few more operations about
00:15:28.000 --> 00:15:31.000
vectors.
And, these operations will be
00:15:31.000 --> 00:15:34.000
useful to us when we start
trying to do a bit of geometry.
00:15:34.000 --> 00:15:37.000
So, of course,
you've all done some geometry.
00:15:37.000 --> 00:15:40.000
But, we are going to see that
geometry can be done using
00:15:40.000 --> 00:15:42.000
vectors.
And, in many ways,
00:15:42.000 --> 00:15:44.000
it's the right language for
that,
00:15:44.000 --> 00:15:47.000
and in particular when we learn
about functions we really will
00:15:47.000 --> 00:15:51.000
want to use vectors more than,
maybe, the other kind of
00:15:51.000 --> 00:15:54.000
geometry that you've seen
before.
00:15:54.000 --> 00:15:56.000
I mean, of course,
it's just a language in a way.
00:15:56.000 --> 00:15:59.000
I mean, we are just
reformulating things that you
00:15:59.000 --> 00:16:02.000
have seen, you already know
since childhood.
00:16:02.000 --> 00:16:07.000
But, you will see that notation
somehow helps to make it more
00:16:07.000 --> 00:16:10.000
straightforward.
So, what is dot product?
00:16:10.000 --> 00:16:16.000
Well, dot product as a way of
multiplying two vectors to get a
00:16:16.000 --> 00:16:21.000
number, a scalar.
And, well, let me start by
00:16:21.000 --> 00:16:25.000
giving you a definition in terms
of components.
00:16:25.000 --> 00:16:29.000
What we do, let's say that we
have a vector,
00:16:29.000 --> 00:16:32.000
A, with components a1,
a2, a3, vector B with
00:16:32.000 --> 00:16:34.000
components b1,
b2, b3.
00:16:34.000 --> 00:16:38.000
Well, we multiply the first
components by the first
00:16:38.000 --> 00:16:43.000
components, the second by the
second, the third by the third.
00:16:43.000 --> 00:16:46.000
If you have N components,
you keep going.
00:16:46.000 --> 00:16:49.000
And, you sum all of these
together.
00:16:49.000 --> 00:16:55.000
OK, and important:
this is a scalar.
00:16:55.000 --> 00:16:59.000
OK, you do not get a vector.
You get a number.
00:16:59.000 --> 00:17:01.000
I know it sounds completely
obvious from the definition
00:17:01.000 --> 00:17:03.000
here,
but in the middle of the action
00:17:03.000 --> 00:17:07.000
when you're going to do
complicated problems,
00:17:07.000 --> 00:17:14.000
it's sometimes easy to forget.
So, that's the definition.
00:17:14.000 --> 00:17:17.000
What is it good for?
Why would we ever want to do
00:17:17.000 --> 00:17:20.000
that?
That's kind of a strange
00:17:20.000 --> 00:17:23.000
operation.
So, probably to see what it's
00:17:23.000 --> 00:17:27.000
good for, I should first tell
you what it is geometrically.
00:17:27.000 --> 00:17:29.000
OK, so what does it do
geometrically?
00:17:38.000 --> 00:17:42.000
Well, what you do when you
multiply two vectors in this
00:17:42.000 --> 00:17:45.000
way,
I claim the answer is equal to
00:17:45.000 --> 00:17:51.000
the length of A times the length
of B times the cosine of the
00:17:51.000 --> 00:17:59.000
angle between them.
So, I have my vector, A,
00:17:59.000 --> 00:18:04.000
and if I have my vector, B,
and I have some angle between
00:18:04.000 --> 00:18:06.000
them,
I multiply the length of A
00:18:06.000 --> 00:18:10.000
times the length of B times the
cosine of that angle.
00:18:10.000 --> 00:18:13.000
So, that looks like a very
artificial operation.
00:18:13.000 --> 00:18:16.000
I mean, why would want to do
that complicated multiplication?
00:18:16.000 --> 00:18:21.000
Well, the basic answer is it
tells us at the same time about
00:18:21.000 --> 00:18:25.000
lengths and about angles.
And, the extra bonus thing is
00:18:25.000 --> 00:18:29.000
that it's very easy to compute
if you have components,
00:18:29.000 --> 00:18:32.000
see, that formula is actually
pretty easy.
00:18:32.000 --> 00:18:39.000
So, OK, maybe I should first
tell you, how do we get this
00:18:39.000 --> 00:18:41.000
from that?
Because, you know,
00:18:41.000 --> 00:18:44.000
in math, one tries to justify
everything to prove theorems.
00:18:44.000 --> 00:18:45.000
So, if you want,
that's the theorem.
00:18:45.000 --> 00:18:47.000
That's the first theorem in
18.02.
00:18:47.000 --> 00:18:52.000
So, how do we prove the theorem?
How do we check that this is,
00:18:52.000 --> 00:18:55.000
indeed, correct using this
definition?
00:18:55.000 --> 00:19:06.000
So, in more common language,
what does this geometric
00:19:06.000 --> 00:19:11.000
definition mean?
Well, the first thing it means,
00:19:11.000 --> 00:19:14.000
before we multiply two vectors,
let's start multiplying a
00:19:14.000 --> 00:19:17.000
vector with itself.
That's probably easier.
00:19:17.000 --> 00:19:19.000
So, if we multiply a vector,
A, with itself,
00:19:19.000 --> 00:19:22.000
using this dot product,
so, by the way,
00:19:22.000 --> 00:19:24.000
I should point out,
we put this dot here.
00:19:24.000 --> 00:19:28.000
That's why it's called dot
product.
00:19:28.000 --> 00:19:33.000
So, what this tells us is we
should get the same thing as
00:19:33.000 --> 00:19:38.000
multiplying the length of A with
itself, so, squared,
00:19:38.000 --> 00:19:43.000
times the cosine of the angle.
But now, the cosine of an
00:19:43.000 --> 00:19:49.000
angle, of zero,
cosine of zero you all know is
00:19:49.000 --> 00:19:52.000
one.
OK, so that's going to be
00:19:52.000 --> 00:19:56.000
length A^2.
Well, doesn't stand a chance of
00:19:56.000 --> 00:19:57.000
being true?
Well, let's see.
00:19:57.000 --> 00:20:03.000
If we do AdotA using this
formula, we will get a1^2 a2^2
00:20:03.000 --> 00:20:07.000
a3^2.
That is, indeed,
00:20:07.000 --> 00:20:14.000
the square of the length.
So, check.
00:20:14.000 --> 00:20:18.000
That works.
OK, now, what about two
00:20:18.000 --> 00:20:23.000
different vectors?
Can we understand what this
00:20:23.000 --> 00:20:27.000
says, and how it relates to
that?
00:20:27.000 --> 00:20:33.000
So, let's say that I have two
different vectors,
00:20:33.000 --> 00:20:40.000
A and B, and I want to try to
understand what's going on.
00:20:40.000 --> 00:20:45.000
So, my claim is that we are
going to be able to understand
00:20:45.000 --> 00:20:49.000
the relation between this and
that in terms of the law of
00:20:49.000 --> 00:20:52.000
cosines.
So, the law of cosines is
00:20:52.000 --> 00:20:56.000
something that tells you about
the length of the third side in
00:20:56.000 --> 00:21:00.000
the triangle like this in terms
of these two sides,
00:21:00.000 --> 00:21:07.000
and the angle here.
OK, so the law of cosines,
00:21:07.000 --> 00:21:11.000
which hopefully you have seen
before, says that,
00:21:11.000 --> 00:21:14.000
so let me give a name to this
side.
00:21:14.000 --> 00:21:19.000
Let's call this side C,
and as a vector,
00:21:19.000 --> 00:21:29.000
C is A minus B.
It's minus B plus A.
00:21:29.000 --> 00:21:37.000
So, it's getting a bit
cluttered here.
00:21:37.000 --> 00:21:45.000
So, the law of cosines says
that the length of the third
00:21:45.000 --> 00:21:53.000
side in this triangle is equal
to length A2 plus length B2.
00:21:53.000 --> 00:21:56.000
Well, if I stopped here,
that would be Pythagoras,
00:21:56.000 --> 00:22:01.000
but I don't have a right angle.
So, I have a third term which
00:22:01.000 --> 00:22:07.000
is twice length A,
length B, cosine theta,
00:22:07.000 --> 00:22:10.000
OK?
Has everyone seen this formula
00:22:10.000 --> 00:22:13.000
sometime?
I hear some yeah's.
00:22:13.000 --> 00:22:16.000
I hear some no's.
Well, it's a fact about,
00:22:16.000 --> 00:22:19.000
I mean, you probably haven't
seen it with vectors,
00:22:19.000 --> 00:22:22.000
but it's a fact about the side
lengths in a triangle.
00:22:22.000 --> 00:22:27.000
And, well, let's say,
if you haven't seen it before,
00:22:27.000 --> 00:22:32.000
then this is going to be a
proof of the law of cosines if
00:22:32.000 --> 00:22:39.000
you believe this.
Otherwise, it's the other way
00:22:39.000 --> 00:22:43.000
around.
So, let's try to see how this
00:22:43.000 --> 00:22:47.000
relates to what I'm saying about
the dot product.
00:22:47.000 --> 00:22:54.000
So, I've been saying that
length C^2, that's the same
00:22:54.000 --> 00:22:56.000
thing as CdotC,
OK?
00:22:56.000 --> 00:23:01.000
That, we have checked.
Now, CdotC, well,
00:23:01.000 --> 00:23:06.000
C is A minus B.
So, it's A minus B,
00:23:06.000 --> 00:23:09.000
dot product,
A minus B.
00:23:09.000 --> 00:23:11.000
Now, what do we want to do in a
situation like that?
00:23:11.000 --> 00:23:16.000
Well, we want to expand this
into a sum of four terms.
00:23:16.000 --> 00:23:19.000
Are we allowed to do that?
Well, we have this dot product
00:23:19.000 --> 00:23:22.000
that's a mysterious new
operation.
00:23:22.000 --> 00:23:24.000
We don't really know.
Well, the answer is yes,
00:23:24.000 --> 00:23:27.000
we can do it.
You can check from this
00:23:27.000 --> 00:23:31.000
definition that it behaves in
the usual way in terms of
00:23:31.000 --> 00:23:34.000
expanding, vectoring,
and so on.
00:23:34.000 --> 00:23:49.000
So, I can write that as AdotA
minus AdotB minus BdotA plus
00:23:49.000 --> 00:23:55.000
BdotB.
So, AdotA is length A^2.
00:23:55.000 --> 00:23:56.000
Let me jump ahead to the last
term.
00:23:56.000 --> 00:24:01.000
BdotB is length B^2,
and then these two terms,
00:24:01.000 --> 00:24:04.000
well, they're the same.
You can check from the
00:24:04.000 --> 00:24:07.000
definition that AdotB and BdotA
are the same thing.
00:24:20.000 --> 00:24:24.000
Well, you see that this term,
I mean, this is the only
00:24:24.000 --> 00:24:30.000
difference between these two
formulas for the length of C.
00:24:30.000 --> 00:24:34.000
So, if you believe in the law
of cosines, then it tells you
00:24:34.000 --> 00:24:39.000
that, yes, this a proof that
AdotB equals length A length B
00:24:39.000 --> 00:24:41.000
cosine theta.
Or, vice versa,
00:24:41.000 --> 00:24:45.000
if you've never seen the law of
cosines, you are willing to
00:24:45.000 --> 00:24:49.000
believe this.
Then, this is the proof of the
00:24:49.000 --> 00:24:53.000
law of cosines.
So, the law of cosines,
00:24:53.000 --> 00:24:59.000
or this interpretation,
are equivalent to each other.
00:24:59.000 --> 00:25:07.000
OK, any questions?
Yes?
00:25:07.000 --> 00:25:12.000
So, in the second one there
isn't a cosine theta because I'm
00:25:12.000 --> 00:25:16.000
just expanding a dot product.
OK, so I'm just writing C
00:25:16.000 --> 00:25:19.000
equals A minus B,
and then I'm expanding this
00:25:19.000 --> 00:25:22.000
algebraically.
And then, I get to an answer
00:25:22.000 --> 00:25:24.000
that has an A.B.
So then, if I wanted to express
00:25:24.000 --> 00:25:27.000
that without a dot product,
then I would have to introduce
00:25:27.000 --> 00:25:31.000
a cosine.
And, I would get the same as
00:25:31.000 --> 00:25:34.000
that, OK?
So, yeah, if you want,
00:25:34.000 --> 00:25:38.000
the next step to recall the law
of cosines would be plug in this
00:25:38.000 --> 00:25:43.000
formula for AdotB.
And then you would have a
00:25:43.000 --> 00:25:58.000
cosine.
OK, let's keep going.
00:25:58.000 --> 00:26:03.000
OK, so what is this good for?
Now that we have a definition,
00:26:03.000 --> 00:26:06.000
we should figure out what we
can do with it.
00:26:06.000 --> 00:26:11.000
So, what are the applications
of dot product?
00:26:11.000 --> 00:26:14.000
Well, will this discover new
applications of dot product
00:26:14.000 --> 00:26:17.000
throughout the entire
semester,but let me tell you at
00:26:17.000 --> 00:26:20.000
least about those that are
readily visible.
00:26:20.000 --> 00:26:33.000
So, one is to compute lengths
and angles, especially angles.
00:26:33.000 --> 00:26:39.000
So, let's do an example.
Let's say that,
00:26:39.000 --> 00:26:44.000
for example,
I have in space,
00:26:44.000 --> 00:26:51.000
I have a point,
P, which is at (1,0,0).
00:26:51.000 --> 00:26:55.000
I have a point,
Q, which is at (0,1,0).
00:26:55.000 --> 00:26:58.000
So, it's at distance one here,
one here.
00:26:58.000 --> 00:27:03.000
And, I have a third point,
R at (0,0,2),
00:27:03.000 --> 00:27:07.000
so it's at height two.
And, let's say that I'm
00:27:07.000 --> 00:27:11.000
curious, and I'm wondering what
is the angle here?
00:27:11.000 --> 00:27:15.000
So, here I have a triangle in
space connect P,
00:27:15.000 --> 00:27:20.000
Q, and R, and I'm wondering,
what is this angle here?
00:27:20.000 --> 00:27:23.000
OK, so, of course,
one solution is to build a
00:27:23.000 --> 00:27:25.000
model and then go and measure
the angle.
00:27:25.000 --> 00:27:28.000
But, we can do better than that.
We can just find the angle
00:27:28.000 --> 00:27:32.000
using dot product.
So, how would we do that?
00:27:32.000 --> 00:27:38.000
Well, so, if we look at this
formula, we see,
00:27:38.000 --> 00:27:44.000
so, let's say that we want to
find the angle here.
00:27:44.000 --> 00:27:50.000
Well, let's look at the formula
for PQdotPR.
00:27:50.000 --> 00:27:56.000
Well, we said it should be
length PQ times length PR times
00:27:56.000 --> 00:27:59.000
the cosine of the angle,
OK?
00:27:59.000 --> 00:28:01.000
Now, what do we know,
and what do we not know?
00:28:01.000 --> 00:28:04.000
Well, certainly at this point
we don't know the cosine of the
00:28:04.000 --> 00:28:06.000
angle.
That's what we would like to
00:28:06.000 --> 00:28:08.000
find.
The lengths,
00:28:08.000 --> 00:28:11.000
certainly we can compute.
We know how to find these
00:28:11.000 --> 00:28:14.000
lengths.
And, this dot product we know
00:28:14.000 --> 00:28:17.000
how to compute because we have
an easy formula here.
00:28:17.000 --> 00:28:20.000
OK, so we can compute
everything else and then find
00:28:20.000 --> 00:28:25.000
theta.
So, I'll tell you what we will
00:28:25.000 --> 00:28:31.000
do is we will find theta -- --
in this way.
00:28:31.000 --> 00:28:34.000
We'll take the dot product of
PQ with PR, and then we'll
00:28:34.000 --> 00:28:36.000
divide by the lengths.
00:29:14.000 --> 00:29:27.000
OK, so let's see.
So, we said cosine theta is
00:29:27.000 --> 00:29:33.000
PQdotPR over length PQ length
PR.
00:29:33.000 --> 00:29:36.000
So, let's try to figure out
what this vector,
00:29:36.000 --> 00:29:39.000
PQ,
well, to go from P to Q,
00:29:39.000 --> 00:29:43.000
I should go minus one unit
along the x direction plus one
00:29:43.000 --> 00:29:46.000
unit along the y direction.
And, I'm not moving in the z
00:29:46.000 --> 00:29:49.000
direction.
So, to go from P to Q,
00:29:49.000 --> 00:29:54.000
I have to move by
***amp***lt;-1,1,0***amp***gt;.
00:29:54.000 --> 00:29:59.000
To go from P to R,
I go -1 along the x axis and 2
00:29:59.000 --> 00:30:04.000
along the z axis.
So, PR, I claim, is this.
00:30:04.000 --> 00:30:12.000
OK, then, the lengths of these
vectors, well,(-1)^2 (1)^2
00:30:12.000 --> 00:30:19.000
(0)^2, square root,
and then same thing with the
00:30:19.000 --> 00:30:24.000
other one.
OK, so, the denominator will
00:30:24.000 --> 00:30:30.000
become the square root of 2,
and there's a square root of 5.
00:30:30.000 --> 00:30:34.000
What about the numerator?
Well, so, remember,
00:30:34.000 --> 00:30:37.000
to do the dot product,
we multiply this by this,
00:30:37.000 --> 00:30:40.000
and that by that,
that by that.
00:30:40.000 --> 00:30:45.000
And, we add.
Minus 1 times minus 1 makes 1
00:30:45.000 --> 00:30:49.000
plus 1 times 0,
that's 0.
00:30:49.000 --> 00:30:55.000
Zero times 2 is 0 again.
So, we will get 1 over square
00:30:55.000 --> 00:30:59.000
root of 10.
That's the cosine of the angle.
00:30:59.000 --> 00:31:03.000
And, of course if we want the
actual angle,
00:31:03.000 --> 00:31:08.000
well, we have to take a
calculator, find the inverse
00:31:08.000 --> 00:31:12.000
cosine, and you'll find it's
about 71.5°.
00:31:12.000 --> 00:31:18.000
Actually, we'll be using mostly
radians, but for today,
00:31:18.000 --> 00:31:26.000
that's certainly more speaking.
OK, any questions about that?
00:31:26.000 --> 00:31:29.000
No?
OK, so in particular,
00:31:29.000 --> 00:31:32.000
I should point out one thing
that's really neat about the
00:31:32.000 --> 00:31:34.000
answer.
I mean, we got this number.
00:31:34.000 --> 00:31:37.000
We don't really know what it
means exactly because it mixes
00:31:37.000 --> 00:31:39.000
together the lengths and the
angle.
00:31:39.000 --> 00:31:41.000
But, one thing that's
interesting here,
00:31:41.000 --> 00:31:45.000
it's the sign of the answer,
the fact that we got a positive
00:31:45.000 --> 00:31:48.000
number.
So, if you think about it,
00:31:48.000 --> 00:31:50.000
the lengths are always
positive.
00:31:50.000 --> 00:31:56.000
So, the sign of a dot product
is the same as a sign of cosine
00:31:56.000 --> 00:32:00.000
theta.
So, in fact,
00:32:00.000 --> 00:32:13.000
the sign of AdotB is going to
be positive if the angle is less
00:32:13.000 --> 00:32:17.000
than 90°.
So, that means geometrically,
00:32:17.000 --> 00:32:21.000
my two vectors are going more
or less in the same direction.
00:32:21.000 --> 00:32:27.000
They make an acute angle.
It's going to be zero if the
00:32:27.000 --> 00:32:33.000
angle is exactly 90°,
OK, because that's when the
00:32:33.000 --> 00:32:39.000
cosine will be zero.
And, it will be negative if the
00:32:39.000 --> 00:32:43.000
angle is more than 90°.
So, that means they go,
00:32:43.000 --> 00:32:46.000
however, in opposite
directions.
00:32:46.000 --> 00:32:50.000
So, that's basically one way to
think about what dot product
00:32:50.000 --> 00:32:54.000
measures.
It measures how much the two
00:32:54.000 --> 00:32:58.000
vectors are going along each
other.
00:32:58.000 --> 00:33:02.000
OK, and that actually leads us
to the next application.
00:33:02.000 --> 00:33:05.000
So, let's see,
did I have a number one there?
00:33:05.000 --> 00:33:07.000
Yes.
So, if I had a number one,
00:33:07.000 --> 00:33:12.000
I must have number two.
The second application is to
00:33:12.000 --> 00:33:16.000
detect orthogonality.
It's to figure out when two
00:33:16.000 --> 00:33:21.000
things are perpendicular.
OK, so orthogonality is just a
00:33:21.000 --> 00:33:26.000
complicated word from Greek to
say things are perpendicular.
00:33:26.000 --> 00:33:34.000
So, let's just take an example.
Let's say I give you the
00:33:34.000 --> 00:33:41.000
equation x 2y 3z = 0.
OK, so that defines a certain
00:33:41.000 --> 00:33:46.000
set of points in space,
and what do you think the set
00:33:46.000 --> 00:33:52.000
of solutions look like if I give
you this equation?
00:33:52.000 --> 00:34:01.000
So far I see one,
two, three answers,
00:34:01.000 --> 00:34:06.000
OK.
So, I see various competing
00:34:06.000 --> 00:34:11.000
answers, but,
yeah, I see a lot of people
00:34:11.000 --> 00:34:18.000
voting for answer number four.
I see also some I don't knows,
00:34:18.000 --> 00:34:22.000
and some other things.
But, the majority vote seems to
00:34:22.000 --> 00:34:26.000
be a plane.
And, indeed that's the correct
00:34:26.000 --> 00:34:28.000
answer.
So, how do we see that it's a
00:34:28.000 --> 00:34:28.000
plane?
00:34:43.000 --> 00:34:49.000
So, I should say,
this is the equation of a
00:34:49.000 --> 00:34:52.000
plane.
So, there's many ways to see
00:34:52.000 --> 00:34:55.000
that, and I'm not going to give
you all of them.
00:34:55.000 --> 00:34:58.000
But, here's one way to think
about it.
00:34:58.000 --> 00:35:03.000
So, let's think geometrically
about how to express this
00:35:03.000 --> 00:35:09.000
condition in terms of vectors.
So, let's take the origin O,
00:35:09.000 --> 00:35:13.000
by convention is the point
(0,0,0).
00:35:13.000 --> 00:35:18.000
And, let's take a point,
P, that will satisfy this
00:35:18.000 --> 00:35:21.000
equation on it,
so, at coordinates x,
00:35:21.000 --> 00:35:24.000
y, z.
So, what does this condition
00:35:24.000 --> 00:35:28.000
here mean?
Well, it means the following
00:35:28.000 --> 00:35:32.000
thing.
So, let's take the vector, OP.
00:35:32.000 --> 00:35:37.000
OK, so vector OP,
of course, has components x,
00:35:37.000 --> 00:35:40.000
y, z.
Now, we can think of this as
00:35:40.000 --> 00:35:44.000
actually a dot product between
OP and a mysterious vector that
00:35:44.000 --> 00:35:47.000
won't remain mysterious for very
long,
00:35:47.000 --> 00:35:50.000
namely, the vector one,
two, three.
00:35:50.000 --> 00:35:59.000
OK, so, this condition is the
same as OP.A equals zero,
00:35:59.000 --> 00:36:03.000
right?
If I take the dot product
00:36:03.000 --> 00:36:09.000
OPdotA I get x times one plus y
times two plus z times three.
00:36:09.000 --> 00:36:14.000
But now, what does it mean that
the dot product between OP and A
00:36:14.000 --> 00:36:19.000
is zero?
Well, it means that OP and A
00:36:19.000 --> 00:36:25.000
are perpendicular.
OK, so I have this vector, A.
00:36:25.000 --> 00:36:28.000
I'm not going to be able to
draw it realistically.
00:36:28.000 --> 00:36:32.000
Let's say it goes this way.
Then, a point,
00:36:32.000 --> 00:36:37.000
P, solves this equation exactly
when the vector from O to P is
00:36:37.000 --> 00:36:40.000
perpendicular to A.
And, I claim that defines a
00:36:40.000 --> 00:36:41.000
plane.
For example,
00:36:41.000 --> 00:36:45.000
if it helps you to see it,
take a vertical vector.
00:36:45.000 --> 00:36:47.000
What does it mean to be
perpendicular to the vertical
00:36:47.000 --> 00:36:49.000
vector?
It means you are horizontal.
00:36:49.000 --> 00:36:56.000
It's the horizontal plane.
Here, it's a plane that passes
00:36:56.000 --> 00:37:05.000
through the origin and is
perpendicular to this vector,
00:37:05.000 --> 00:37:14.000
A.
OK, so what we get is a plane
00:37:14.000 --> 00:37:25.000
through the origin perpendicular
to A.
00:37:25.000 --> 00:37:29.000
And, in general,
what you should remember is
00:37:29.000 --> 00:37:35.000
that two vectors have a dot
product equal to zero if and
00:37:35.000 --> 00:37:41.000
only if that's equivalent to the
cosine of the angle between them
00:37:41.000 --> 00:37:46.000
is zero.
That means the angle is 90°.
00:37:46.000 --> 00:37:51.000
That means A and B are
perpendicular.
00:37:51.000 --> 00:37:57.000
So, we have a very fast way of
checking whether two vectors are
00:37:57.000 --> 00:38:01.000
perpendicular.
So, one additional application
00:38:01.000 --> 00:38:05.000
I think we'll see actually
tomorrow is to find the
00:38:05.000 --> 00:38:10.000
components of a vector along a
certain direction.
00:38:10.000 --> 00:38:13.000
So, I claim we can use this
intuition I gave about dot
00:38:13.000 --> 00:38:16.000
product telling us how much to
vectors go in the same direction
00:38:16.000 --> 00:38:19.000
to actually give a precise
meaning to the notion of
00:38:19.000 --> 00:38:22.000
component for vector,
not just along the x,
00:38:22.000 --> 00:38:27.000
y, or z axis,
but along any direction in
00:38:27.000 --> 00:38:31.000
space.
So, I think I should probably
00:38:31.000 --> 00:38:34.000
stop here.
But, I will see you tomorrow at
00:38:34.000 --> 00:38:38.000
2:00 here, and we'll learn more
about that and about cross
00:38:38.000 --> 00:38:44.000
products.