jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

video_id stringclasses 7 values	text stringlengths 2 29.3k
PxCxlsl_YwY	using this dot product, so, by the way,
PxCxlsl_YwY	I should point out, we put this dot here.
PxCxlsl_YwY	That's why it's called dot product.
PxCxlsl_YwY	So, what this tells us is we should get the same thing as
PxCxlsl_YwY	multiplying the length of A with itself, so, squared,
PxCxlsl_YwY	times the cosine of the angle. But now, the cosine of an
PxCxlsl_YwY	angle, of zero, cosine of zero you all know is
PxCxlsl_YwY	one. OK, so that's going to be
PxCxlsl_YwY	length A^2. Well, doesn't stand a chance of
PxCxlsl_YwY	being true? Well, let's see.
PxCxlsl_YwY	If we do AdotA using this formula, we will get a1^2 a2^2
PxCxlsl_YwY	a3^2. That is, indeed,
PxCxlsl_YwY	the square of the length. So, check.
PxCxlsl_YwY	That works. OK, now, what about two
PxCxlsl_YwY	different vectors? Can we understand what this
PxCxlsl_YwY	says, and how it relates to that?
PxCxlsl_YwY	So, let's say that I have two different vectors,
PxCxlsl_YwY	A and B, and I want to try to understand what's going on.
PxCxlsl_YwY	So, my claim is that we are going to be able to understand
PxCxlsl_YwY	the relation between this and that in terms of the law of
PxCxlsl_YwY	cosines. So, the law of cosines is
PxCxlsl_YwY	something that tells you about the length of the third side in
PxCxlsl_YwY	the triangle like this in terms of these two sides,
PxCxlsl_YwY	and the angle here. OK, so the law of cosines,
PxCxlsl_YwY	which hopefully you have seen before, says that,
PxCxlsl_YwY	so let me give a name to this side.
PxCxlsl_YwY	Let's call this side C, and as a vector,
PxCxlsl_YwY	C is A minus B. It's minus B plus A.
PxCxlsl_YwY	So, it's getting a bit cluttered here.
PxCxlsl_YwY	So, the law of cosines says that the length of the third
PxCxlsl_YwY	side in this triangle is equal to length A2 plus length B2.
PxCxlsl_YwY	Well, if I stopped here, that would be Pythagoras,
PxCxlsl_YwY	but I don't have a right angle. So, I have a third term which
PxCxlsl_YwY	is twice length A, length B, cosine theta,
PxCxlsl_YwY	OK? Has everyone seen this formula
PxCxlsl_YwY	sometime? I hear some yeah's.
PxCxlsl_YwY	I hear some no's. Well, it's a fact about,
PxCxlsl_YwY	I mean, you probably haven't seen it with vectors,
PxCxlsl_YwY	but it's a fact about the side lengths in a triangle.
PxCxlsl_YwY	And, well, let's say, if you haven't seen it before,
PxCxlsl_YwY	then this is going to be a proof of the law of cosines if
PxCxlsl_YwY	you believe this. Otherwise, it's the other way
PxCxlsl_YwY	around. So, let's try to see how this
PxCxlsl_YwY	relates to what I'm saying about the dot product.
PxCxlsl_YwY	So, I've been saying that length C^2, that's the same
PxCxlsl_YwY	thing as CdotC, OK?
PxCxlsl_YwY	That, we have checked. Now, CdotC, well,
PxCxlsl_YwY	C is A minus B. So, it's A minus B,
PxCxlsl_YwY	dot product, A minus B.
PxCxlsl_YwY	Now, what do we want to do in a situation like that?
PxCxlsl_YwY	Well, we want to expand this into a sum of four terms.
PxCxlsl_YwY	Are we allowed to do that? Well, we have this dot product
PxCxlsl_YwY	that's a mysterious new operation.
PxCxlsl_YwY	We don't really know. Well, the answer is yes,
PxCxlsl_YwY	we can do it. You can check from this
PxCxlsl_YwY	definition that it behaves in the usual way in terms of
PxCxlsl_YwY	expanding, vectoring, and so on.
PxCxlsl_YwY	So, I can write that as AdotA minus AdotB minus BdotA plus
PxCxlsl_YwY	BdotB. So, AdotA is length A^2.
PxCxlsl_YwY	Let me jump ahead to the last term.
PxCxlsl_YwY	BdotB is length B^2, and then these two terms,
PxCxlsl_YwY	well, they're the same. You can check from the
PxCxlsl_YwY	definition that AdotB and BdotA are the same thing.
PxCxlsl_YwY	Well, you see that this term, I mean, this is the only
PxCxlsl_YwY	difference between these two formulas for the length of C.
PxCxlsl_YwY	So, if you believe in the law of cosines, then it tells you
PxCxlsl_YwY	that, yes, this a proof that AdotB equals length A length B
PxCxlsl_YwY	cosine theta. Or, vice versa,
PxCxlsl_YwY	if you've never seen the law of cosines, you are willing to
PxCxlsl_YwY	believe this. Then, this is the proof of the
PxCxlsl_YwY	law of cosines. So, the law of cosines,
PxCxlsl_YwY	or this interpretation, are equivalent to each other.
PxCxlsl_YwY	OK, any questions? Yes?
PxCxlsl_YwY	So, in the second one there isn't a cosine theta because I'm
PxCxlsl_YwY	just expanding a dot product. OK, so I'm just writing C
PxCxlsl_YwY	equals A minus B, and then I'm expanding this
PxCxlsl_YwY	algebraically. And then, I get to an answer
PxCxlsl_YwY	that has an A.B. So then, if I wanted to express
PxCxlsl_YwY	that without a dot product, then I would have to introduce
PxCxlsl_YwY	a cosine. And, I would get the same as
PxCxlsl_YwY	that, OK? So, yeah, if you want,
PxCxlsl_YwY	the next step to recall the law of cosines would be plug in this
PxCxlsl_YwY	formula for AdotB. And then you would have a
PxCxlsl_YwY	cosine. OK, let's keep going.
PxCxlsl_YwY	OK, so what is this good for? Now that we have a definition,
PxCxlsl_YwY	we should figure out what we can do with it.
PxCxlsl_YwY	So, what are the applications of dot product?
PxCxlsl_YwY	Well, will this discover new applications of dot product
PxCxlsl_YwY	throughout the entire semester,but let me tell you at
PxCxlsl_YwY	least about those that are readily visible.
PxCxlsl_YwY	So, one is to compute lengths and angles, especially angles.
PxCxlsl_YwY	So, let's do an example. Let's say that,
PxCxlsl_YwY	for example, I have in space,
PxCxlsl_YwY	I have a point, P, which is at (1,0,0).
PxCxlsl_YwY	I have a point, Q, which is at (0,1,0).
PxCxlsl_YwY	So, it's at distance one here, one here.
PxCxlsl_YwY	And, I have a third point, R at (0,0,2),
PxCxlsl_YwY	so it's at height two. And, let's say that I'm
PxCxlsl_YwY	curious, and I'm wondering what is the angle here?
PxCxlsl_YwY	So, here I have a triangle in space connect P,

PxCxlsl_YwY

using this dot product, so, by the way,

PxCxlsl_YwY

I should point out, we put this dot here.

PxCxlsl_YwY

That's why it's called dot product.

PxCxlsl_YwY

So, what this tells us is we should get the same thing as

PxCxlsl_YwY

multiplying the length of A with itself, so, squared,

PxCxlsl_YwY

times the cosine of the angle. But now, the cosine of an

PxCxlsl_YwY

angle, of zero, cosine of zero you all know is

PxCxlsl_YwY

one. OK, so that's going to be

PxCxlsl_YwY

length A^2. Well, doesn't stand a chance of

PxCxlsl_YwY

being true? Well, let's see.

PxCxlsl_YwY

If we do AdotA using this formula, we will get a1^2 a2^2

PxCxlsl_YwY

a3^2. That is, indeed,

PxCxlsl_YwY

the square of the length. So, check.

PxCxlsl_YwY

That works. OK, now, what about two

PxCxlsl_YwY

different vectors? Can we understand what this

PxCxlsl_YwY

says, and how it relates to that?

PxCxlsl_YwY

So, let's say that I have two different vectors,

PxCxlsl_YwY

A and B, and I want to try to understand what's going on.

PxCxlsl_YwY

So, my claim is that we are going to be able to understand

PxCxlsl_YwY

the relation between this and that in terms of the law of

PxCxlsl_YwY

cosines. So, the law of cosines is

PxCxlsl_YwY

something that tells you about the length of the third side in

PxCxlsl_YwY

the triangle like this in terms of these two sides,

PxCxlsl_YwY

and the angle here. OK, so the law of cosines,

PxCxlsl_YwY

which hopefully you have seen before, says that,

PxCxlsl_YwY

so let me give a name to this side.

PxCxlsl_YwY

Let's call this side C, and as a vector,

PxCxlsl_YwY

C is A minus B. It's minus B plus A.

PxCxlsl_YwY

So, it's getting a bit cluttered here.

PxCxlsl_YwY

So, the law of cosines says that the length of the third

PxCxlsl_YwY

side in this triangle is equal to length A2 plus length B2.

PxCxlsl_YwY

Well, if I stopped here, that would be Pythagoras,

PxCxlsl_YwY

but I don't have a right angle. So, I have a third term which

PxCxlsl_YwY

is twice length A, length B, cosine theta,

PxCxlsl_YwY

OK? Has everyone seen this formula

PxCxlsl_YwY

sometime? I hear some yeah's.

PxCxlsl_YwY

I hear some no's. Well, it's a fact about,

PxCxlsl_YwY

I mean, you probably haven't seen it with vectors,

PxCxlsl_YwY

but it's a fact about the side lengths in a triangle.

PxCxlsl_YwY

And, well, let's say, if you haven't seen it before,

PxCxlsl_YwY

then this is going to be a proof of the law of cosines if

PxCxlsl_YwY

you believe this. Otherwise, it's the other way

PxCxlsl_YwY

around. So, let's try to see how this

PxCxlsl_YwY

relates to what I'm saying about the dot product.

PxCxlsl_YwY

So, I've been saying that length C^2, that's the same

PxCxlsl_YwY

thing as CdotC, OK?

PxCxlsl_YwY

That, we have checked. Now, CdotC, well,

PxCxlsl_YwY

C is A minus B. So, it's A minus B,

PxCxlsl_YwY

dot product, A minus B.

PxCxlsl_YwY

Now, what do we want to do in a situation like that?

PxCxlsl_YwY

Well, we want to expand this into a sum of four terms.

PxCxlsl_YwY

Are we allowed to do that? Well, we have this dot product

PxCxlsl_YwY

that's a mysterious new operation.

PxCxlsl_YwY

We don't really know. Well, the answer is yes,

PxCxlsl_YwY

we can do it. You can check from this

PxCxlsl_YwY

definition that it behaves in the usual way in terms of

PxCxlsl_YwY

expanding, vectoring, and so on.

PxCxlsl_YwY

So, I can write that as AdotA minus AdotB minus BdotA plus

PxCxlsl_YwY

BdotB. So, AdotA is length A^2.

PxCxlsl_YwY

Let me jump ahead to the last term.

PxCxlsl_YwY

BdotB is length B^2, and then these two terms,

PxCxlsl_YwY

well, they're the same. You can check from the

PxCxlsl_YwY

definition that AdotB and BdotA are the same thing.

PxCxlsl_YwY

Well, you see that this term, I mean, this is the only

PxCxlsl_YwY

difference between these two formulas for the length of C.

PxCxlsl_YwY

So, if you believe in the law of cosines, then it tells you

PxCxlsl_YwY

that, yes, this a proof that AdotB equals length A length B

PxCxlsl_YwY

cosine theta. Or, vice versa,

PxCxlsl_YwY

if you've never seen the law of cosines, you are willing to

PxCxlsl_YwY

believe this. Then, this is the proof of the

PxCxlsl_YwY

law of cosines. So, the law of cosines,

PxCxlsl_YwY

or this interpretation, are equivalent to each other.

PxCxlsl_YwY

OK, any questions? Yes?

PxCxlsl_YwY

So, in the second one there isn't a cosine theta because I'm

PxCxlsl_YwY

just expanding a dot product. OK, so I'm just writing C

PxCxlsl_YwY

equals A minus B, and then I'm expanding this

PxCxlsl_YwY

algebraically. And then, I get to an answer

PxCxlsl_YwY

that has an A.B. So then, if I wanted to express

PxCxlsl_YwY

that without a dot product, then I would have to introduce

PxCxlsl_YwY

a cosine. And, I would get the same as

PxCxlsl_YwY

that, OK? So, yeah, if you want,

PxCxlsl_YwY

the next step to recall the law of cosines would be plug in this

PxCxlsl_YwY

formula for AdotB. And then you would have a

PxCxlsl_YwY

cosine. OK, let's keep going.

PxCxlsl_YwY

OK, so what is this good for? Now that we have a definition,

PxCxlsl_YwY

we should figure out what we can do with it.

PxCxlsl_YwY

So, what are the applications of dot product?

PxCxlsl_YwY

Well, will this discover new applications of dot product

PxCxlsl_YwY

throughout the entire semester,but let me tell you at

PxCxlsl_YwY

least about those that are readily visible.

PxCxlsl_YwY

So, one is to compute lengths and angles, especially angles.

PxCxlsl_YwY

So, let's do an example. Let's say that,

PxCxlsl_YwY

for example, I have in space,

PxCxlsl_YwY

I have a point, P, which is at (1,0,0).

PxCxlsl_YwY

I have a point, Q, which is at (0,1,0).

PxCxlsl_YwY

So, it's at distance one here, one here.

PxCxlsl_YwY

And, I have a third point, R at (0,0,2),

PxCxlsl_YwY

so it's at height two. And, let's say that I'm

PxCxlsl_YwY

curious, and I'm wondering what is the angle here?

PxCxlsl_YwY

So, here I have a triangle in space connect P,