jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

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PxCxlsl_YwY	the first component of B, the second with the second,
PxCxlsl_YwY	and the third with the third. OK, say that A was
PxCxlsl_YwY	*amplt;a1, a2, a3amp*gt;
PxCxlsl_YwY	B was *amplt;b1, b2, b3amp*gt;,
PxCxlsl_YwY	then you just add this way. OK, so it's pretty
PxCxlsl_YwY	straightforward. So, for example,
PxCxlsl_YwY	I said that my vector over there, its components are three,
PxCxlsl_YwY	two, one. But, I also wrote it as 3i 2j k.
PxCxlsl_YwY	What does that mean? OK, so I need to tell you first
PxCxlsl_YwY	about multiplying by a scalar. So, this is about addition.
PxCxlsl_YwY	So, multiplication by a scalar, it's very easy.
PxCxlsl_YwY	If you have a vector, A, then you can form a vector
PxCxlsl_YwY	2A just by making it go twice as far in the same direction.
PxCxlsl_YwY	Or, we can make half A more modestly.
PxCxlsl_YwY	We can even make minus A, and so on.
PxCxlsl_YwY	So now, you see, if I do the calculation,
PxCxlsl_YwY	3i 2j k, well, what does it mean?
PxCxlsl_YwY	3i is just going to go along the x axis, but by distance of
PxCxlsl_YwY	three instead of one. And then, 2j goes two units
PxCxlsl_YwY	along the y axis, and k goes up by one unit.
PxCxlsl_YwY	Well, if you add these together, you will go from the
PxCxlsl_YwY	origin, then along the x axis, then parallel to the y axis,
PxCxlsl_YwY	and then up. And, you will end up,
PxCxlsl_YwY	indeed, at the endpoint of a vector.
PxCxlsl_YwY	OK, any questions at this point? Yes?
PxCxlsl_YwY	Exactly. To add vectors geometrically,
PxCxlsl_YwY	you just put the head of the first vector and the tail of the
PxCxlsl_YwY	second vector in the same place. And then, it's head to tail
PxCxlsl_YwY	addition. Any other questions?
PxCxlsl_YwY	Yes? That's correct.
PxCxlsl_YwY	If you subtract two vectors, that just means you add the
PxCxlsl_YwY	opposite of a vector. So, for example,
PxCxlsl_YwY	if I wanted to do A minus B, I would first go along A and
PxCxlsl_YwY	then along minus B, which would take me somewhere
PxCxlsl_YwY	over there, OK? So, A minus B,
PxCxlsl_YwY	if you want, would go from here to here.
PxCxlsl_YwY	OK, so hopefully you've kind of seen that stuff either before in
PxCxlsl_YwY	your lives, or at least yesterday.
PxCxlsl_YwY	So, I'm going to use that as an excuse to move quickly forward.
PxCxlsl_YwY	So, now we are going to learn a few more operations about
PxCxlsl_YwY	vectors. And, these operations will be
PxCxlsl_YwY	useful to us when we start trying to do a bit of geometry.
PxCxlsl_YwY	So, of course, you've all done some geometry.
PxCxlsl_YwY	But, we are going to see that geometry can be done using
PxCxlsl_YwY	vectors. And, in many ways,
PxCxlsl_YwY	it's the right language for that,
PxCxlsl_YwY	and in particular when we learn about functions we really will
PxCxlsl_YwY	want to use vectors more than, maybe, the other kind of
PxCxlsl_YwY	geometry that you've seen before.
PxCxlsl_YwY	I mean, of course, it's just a language in a way.
PxCxlsl_YwY	I mean, we are just reformulating things that you
PxCxlsl_YwY	have seen, you already know since childhood.
PxCxlsl_YwY	But, you will see that notation somehow helps to make it more
PxCxlsl_YwY	straightforward. So, what is dot product?
PxCxlsl_YwY	Well, dot product as a way of multiplying two vectors to get a
PxCxlsl_YwY	number, a scalar. And, well, let me start by
PxCxlsl_YwY	giving you a definition in terms of components.
PxCxlsl_YwY	What we do, let's say that we have a vector,
PxCxlsl_YwY	A, with components a1, a2, a3, vector B with
PxCxlsl_YwY	components b1, b2, b3.
PxCxlsl_YwY	Well, we multiply the first components by the first
PxCxlsl_YwY	components, the second by the second, the third by the third.
PxCxlsl_YwY	If you have N components, you keep going.
PxCxlsl_YwY	And, you sum all of these together.
PxCxlsl_YwY	OK, and important: this is a scalar.
PxCxlsl_YwY	OK, you do not get a vector. You get a number.
PxCxlsl_YwY	I know it sounds completely obvious from the definition
PxCxlsl_YwY	here, but in the middle of the action
PxCxlsl_YwY	when you're going to do complicated problems,
PxCxlsl_YwY	it's sometimes easy to forget. So, that's the definition.
PxCxlsl_YwY	What is it good for? Why would we ever want to do
PxCxlsl_YwY	that? That's kind of a strange
PxCxlsl_YwY	operation. So, probably to see what it's
PxCxlsl_YwY	good for, I should first tell you what it is geometrically.
PxCxlsl_YwY	OK, so what does it do geometrically?
PxCxlsl_YwY	Well, what you do when you multiply two vectors in this
PxCxlsl_YwY	way, I claim the answer is equal to
PxCxlsl_YwY	the length of A times the length of B times the cosine of the
PxCxlsl_YwY	angle between them. So, I have my vector, A,
PxCxlsl_YwY	and if I have my vector, B, and I have some angle between
PxCxlsl_YwY	them, I multiply the length of A
PxCxlsl_YwY	times the length of B times the cosine of that angle.
PxCxlsl_YwY	So, that looks like a very artificial operation.
PxCxlsl_YwY	I mean, why would want to do that complicated multiplication?
PxCxlsl_YwY	Well, the basic answer is it tells us at the same time about
PxCxlsl_YwY	lengths and about angles. And, the extra bonus thing is
PxCxlsl_YwY	that it's very easy to compute if you have components,
PxCxlsl_YwY	see, that formula is actually pretty easy.
PxCxlsl_YwY	So, OK, maybe I should first tell you, how do we get this
PxCxlsl_YwY	from that? Because, you know,
PxCxlsl_YwY	in math, one tries to justify everything to prove theorems.
PxCxlsl_YwY	So, if you want, that's the theorem.
PxCxlsl_YwY	That's the first theorem in 18.02.
PxCxlsl_YwY	So, how do we prove the theorem? How do we check that this is,
PxCxlsl_YwY	indeed, correct using this definition?
PxCxlsl_YwY	So, in more common language, what does this geometric
PxCxlsl_YwY	definition mean? Well, the first thing it means,
PxCxlsl_YwY	before we multiply two vectors, let's start multiplying a
PxCxlsl_YwY	vector with itself. That's probably easier.
PxCxlsl_YwY	So, if we multiply a vector, A, with itself,

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

straightforward. So, for example,

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

Or, we can make half A more modestly.

PxCxlsl_YwY

We can even make minus A, and so on.

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

and then up. And, you will end up,

PxCxlsl_YwY

indeed, at the endpoint of a vector.

PxCxlsl_YwY

OK, any questions at this point? Yes?

PxCxlsl_YwY

Exactly. To add vectors geometrically,

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

addition. Any other questions?

PxCxlsl_YwY

Yes? That's correct.

PxCxlsl_YwY

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

PxCxlsl_YwY

opposite of a vector. So, for example,

PxCxlsl_YwY

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

PxCxlsl_YwY

then along minus B, which would take me somewhere

PxCxlsl_YwY

over there, OK? So, A minus B,

PxCxlsl_YwY

if you want, would go from here to here.

PxCxlsl_YwY

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

PxCxlsl_YwY

your lives, or at least yesterday.

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

vectors. And, these operations will be

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

vectors. And, in many ways,

PxCxlsl_YwY

it's the right language for that,

PxCxlsl_YwY

and in particular when we learn about functions we really will

PxCxlsl_YwY

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

PxCxlsl_YwY

geometry that you've seen before.

PxCxlsl_YwY

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

PxCxlsl_YwY

I mean, we are just reformulating things that you

PxCxlsl_YwY

have seen, you already know since childhood.

PxCxlsl_YwY

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

PxCxlsl_YwY

straightforward. So, what is dot product?

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

giving you a definition in terms of components.

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

components b1, b2, b3.

PxCxlsl_YwY

Well, we multiply the first components by the first

PxCxlsl_YwY

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

PxCxlsl_YwY

If you have N components, you keep going.

PxCxlsl_YwY

And, you sum all of these together.

PxCxlsl_YwY

OK, and important: this is a scalar.

PxCxlsl_YwY

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

PxCxlsl_YwY

I know it sounds completely obvious from the definition

PxCxlsl_YwY

here, but in the middle of the action

PxCxlsl_YwY

when you're going to do complicated problems,

PxCxlsl_YwY

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

PxCxlsl_YwY

What is it good for? Why would we ever want to do

PxCxlsl_YwY

that? That's kind of a strange

PxCxlsl_YwY

operation. So, probably to see what it's

PxCxlsl_YwY

good for, I should first tell you what it is geometrically.

PxCxlsl_YwY

OK, so what does it do geometrically?

PxCxlsl_YwY

Well, what you do when you multiply two vectors in this

PxCxlsl_YwY

way, I claim the answer is equal to

PxCxlsl_YwY

the length of A times the length of B times the cosine of the

PxCxlsl_YwY

angle between them. So, I have my vector, A,

PxCxlsl_YwY

and if I have my vector, B, and I have some angle between

PxCxlsl_YwY

them, I multiply the length of A

PxCxlsl_YwY

times the length of B times the cosine of that angle.

PxCxlsl_YwY

So, that looks like a very artificial operation.

PxCxlsl_YwY

I mean, why would want to do that complicated multiplication?

PxCxlsl_YwY

Well, the basic answer is it tells us at the same time about

PxCxlsl_YwY

lengths and about angles. And, the extra bonus thing is

PxCxlsl_YwY

that it's very easy to compute if you have components,

PxCxlsl_YwY

see, that formula is actually pretty easy.

PxCxlsl_YwY

So, OK, maybe I should first tell you, how do we get this

PxCxlsl_YwY

from that? Because, you know,

PxCxlsl_YwY

in math, one tries to justify everything to prove theorems.

PxCxlsl_YwY

So, if you want, that's the theorem.

PxCxlsl_YwY

That's the first theorem in 18.02.

PxCxlsl_YwY

So, how do we prove the theorem? How do we check that this is,

PxCxlsl_YwY

indeed, correct using this definition?

PxCxlsl_YwY

So, in more common language, what does this geometric

PxCxlsl_YwY

definition mean? Well, the first thing it means,

PxCxlsl_YwY

before we multiply two vectors, let's start multiplying a

PxCxlsl_YwY

vector with itself. That's probably easier.

PxCxlsl_YwY

So, if we multiply a vector, A, with itself,