before we can go on to more advanced things.

So, hopefully yesterday in recitation, you heard a bit

about vectors. How many of you actually knew

about vectors before that? OK, that's the vast majority.

If you are not one of those people, well,

hopefully you'll learn about vectors right now.

I'm sorry that the learning curve will be a bit steeper for

the first week. But hopefully,

you'll adjust fine. If you have trouble with

vectors, do go to your recitation instructor's office

hours for extra practice if you feel the need to.

You will see it's pretty easy. So, just to remind you,

a vector is a quantity that has both a direction and a magnitude

of length.

So -- So, concretely the way you draw a vector is by some

arrow, like that, OK?

And so, it has a length, and it's pointing in some

direction. And, so, now,

the way that we compute things with vectors,

typically, as we introduce a coordinate system.

So, if we are in the plane, x-y-axis, if we are in space,

x-y-z axis. So, usually I will try to draw

my x-y-z axis consistently to look like this.

And then, I can represent my vector in terms of its

components along the coordinate axis.

So, that means when I have this row, I can ask,

how much does it go in the x direction?

How much does it go in the y direction?

How much does it go in the z direction?

And, so, let's call this a vector A.

So, it's more convention. When we have a vector quantity,

we put an arrow on top to remind us that it's a vector.

If it's in the textbook, then sometimes it's in bold

because it's easier to typeset. If you've tried in your

favorite word processor, bold is easy and vectors are

not easy. So, the vector you can try to

decompose terms of unit vectors directed along the coordinate

axis. So, the convention is there is

a vector that we call ***amp***lt;i***amp***gt;

hat that points along the x axis and has length one.

There's a vector called ***amp***lt;j***amp***gt;

hat that does the same along the y axis,

and the ***amp***lt;k***amp***gt;

hat that does the same along the z axis.

And, so, we can express any vector in terms of its

components. So, the other notation is

***amp***lt;a1, a2, a3 ***amp***gt;

between these square brackets. Well, in angular brackets.

So, the length of a vector we denote by, if you want,

it's the same notation as the absolute value.

So, that's going to be a number, as we say,

now, a scalar quantity. OK, so, a scalar quantity is a

usual numerical quantity as opposed to a vector quantity.

And, its direction is sometimes called dir A,

and that can be obtained just by scaling the vector down to

unit length, for example,

by dividing it by its length. So -- Well, there's a lot of

notation to be learned. So, for example,

if I have two points, P and Q, then I can draw a

vector from P to Q. And, that vector is called

vector PQ, OK? So, maybe we'll call it A.

But, a vector doesn't really have, necessarily,

a starting point and an ending point.

OK, so if I decide to start here and I go by the same

distance in the same direction, this is also vector A.

It's the same thing. So, a lot of vectors we'll draw

starting at the origin, but we don't have to.

So, let's just check and see how things went in recitation.

So, let's say that I give you the vector

***amp***lt;3,2,1***amp***gt;. And so, what do you think about

the length of this vector? OK, I see an answer forming.

So, a lot of you are answering the same thing.

Maybe it shouldn't spoil it for those who haven't given it yet.

OK, I think the overwhelming vote is in favor of answer

number two. I see some sixes, I don't know.

That's a perfectly good answer, too, but hopefully in a few

minutes it won't be I don't know anymore.

So, let's see. How do we find -- -- the length

of a vector three, two, one?

Well, so, this vector, A, it comes towards us along

the x axis by three units. It goes to the right along the

y axis by two units, and then it goes up by one unit

along the z axis. OK, so, it's pointing towards

here. That's pretty hard to draw.

So, how do we get its length? Well, maybe we can start with

something easier, the length of the vector in the

plane. So, observe that A is obtained

from a vector, B, in the plane.

Say, B equals three (i) hat plus two (j) hat.

And then, we just have to, still, go up by one unit,

OK? So, let me try to draw a

picture in this vertical plane that contains A and B.

If I draw it in the vertical plane,

so, that's the Z axis, that's not any particular axis,

then my vector B will go here, and my vector A will go above

it. And here, that's one unit.

And, here I have a right angle. So, I can use the Pythagorean

theorem to find that length A^2 equals length B^2 plus one.

Now, we are reduced to finding the length of B.

The length of B, we can again find using the

Pythagorean theorem in the XY plane because here we have the

right angle. Here we have three units,

and here we have two units. OK, so, if you do the

calculations, you will see that,

well, length of B is square root of (3^2 2^2),

that's 13. So, the square root of 13 -- --

and length of A is square root of length B^2 plus one (square

it if you want) which is going to be square root of 13 plus one

is the square root of 14, hence, answer number two which

almost all of you gave. OK, so the general formula,

if you follow it with it, in general if we have a vector

with components a1, a2, a3,

then the length of A is the square root of a1^2 plus a2^2

plus a3^2. OK, any questions about that?

Yes? Yes.

So, in general, we indeed can consider vectors

in abstract spaces that have any number of coordinates.

And that you have more components.

In this class, we'll mostly see vectors with

two or three components because they are easier to draw,

and because a lot of the math that we'll see works exactly the

same way whether you have three variables or a million

variables. If we had a factor with more

components, then we would have a lot of trouble drawing it.

But we could still define its length in the same way,

by summing the squares of the components.

So, I'm sorry to say that here, multi-variable,

multi will mean mostly two or three.

But, be assured that it works just the same way if you have

10,000 variables. Just, calculations are longer.

OK, more questions? So, what else can we do with

vectors? Well, another thing that I'm

sure you know how to do with vectors is to add them to scale

them. So, vector addition,

so, if you have two vectors, A and B, then you can form,

their sum, A plus B. How do we do that?

Well, first, I should tell you,

vectors, they have this double life.

They are, at the same time, geometric objects that we can

draw like this in pictures, and there are also

computational objects that we can represent by numbers.

So, every question about vectors will have two answers,

one geometric, and one numerical.

OK, so let's start with the geometric.

So, let's say that I have two vectors, A and B,

given to me. And, let's say that I thought

of drawing them at the same place to start with.

Well, to take the sum, what I should do is actually

move B so that it starts at the end of A, at the head of A.

OK, so this is, again, vector B. So, observe,

this actually forms, now, a parallelogram,

right? So, this side is,

again, vector A. And now, if we take the

diagonal of that parallelogram, this is what we call A plus B,

OK, so, the idea being that to move along A plus B,

it's the same as to move first along A and then along B,

or, along B, then along A. A plus B equals B plus A.

OK, now, if we do it numerically,

then all you do is you just add the first component of A with

the first component of B, the second with the second,

and the third with the third. OK, say that A was

***amp***lt;a1, a2, a3***amp***gt;

B was ***amp***lt;b1, b2, b3***amp***gt;,

then you just add this way. OK, so it's pretty

straightforward. So, for example,

I said that my vector over there, its components are three,

two, one. But, I also wrote it as 3i 2j k.

What does that mean? OK, so I need to tell you first

about multiplying by a scalar. So, this is about addition.

So, multiplication by a scalar, it's very easy.

If you have a vector, A, then you can form a vector

2A just by making it go twice as far in the same direction.

Or, we can make half A more modestly.

We can even make minus A, and so on.

So now, you see, if I do the calculation,

3i 2j k, well, what does it mean?

3i is just going to go along the x axis, but by distance of

three instead of one. And then, 2j goes two units

along the y axis, and k goes up by one unit.

Well, if you add these together, you will go from the

origin, then along the x axis, then parallel to the y axis,

and then up. And, you will end up,

indeed, at the endpoint of a vector.

OK, any questions at this point? Yes?

Exactly. To add vectors geometrically,

you just put the head of the first vector and the tail of the

second vector in the same place. And then, it's head to tail

addition. Any other questions?

Yes? That's correct.

If you subtract two vectors, that just means you add the

opposite of a vector. So, for example,

if I wanted to do A minus B, I would first go along A and

then along minus B, which would take me somewhere

over there, OK? So, A minus B,

if you want, would go from here to here.

OK, so hopefully you've kind of seen that stuff either before in

your lives, or at least yesterday.

So, I'm going to use that as an excuse to move quickly forward.

So, now we are going to learn a few more operations about

vectors. And, these operations will be

useful to us when we start trying to do a bit of geometry.

So, of course, you've all done some geometry.

But, we are going to see that geometry can be done using

vectors. And, in many ways,

it's the right language for that,

and in particular when we learn about functions we really will

want to use vectors more than, maybe, the other kind of

geometry that you've seen before.

I mean, of course, it's just a language in a way.

I mean, we are just reformulating things that you

have seen, you already know since childhood.

But, you will see that notation somehow helps to make it more

straightforward. So, what is dot product?

Well, dot product as a way of multiplying two vectors to get a

number, a scalar. And, well, let me start by

giving you a definition in terms of components.

What we do, let's say that we have a vector,

A, with components a1, a2, a3, vector B with

components b1, b2, b3.

Well, we multiply the first components by the first

components, the second by the second, the third by the third.

If you have N components, you keep going.

And, you sum all of these together.

OK, and important: this is a scalar.

OK, you do not get a vector. You get a number.

I know it sounds completely obvious from the definition

here, but in the middle of the action

when you're going to do complicated problems,

it's sometimes easy to forget. So, that's the definition.