jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

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PxCxlsl_YwY	vector from P to Q. And, that vector is called
PxCxlsl_YwY	vector PQ, OK? So, maybe we'll call it A.
PxCxlsl_YwY	But, a vector doesn't really have, necessarily,
PxCxlsl_YwY	a starting point and an ending point.
PxCxlsl_YwY	OK, so if I decide to start here and I go by the same
PxCxlsl_YwY	distance in the same direction, this is also vector A.
PxCxlsl_YwY	It's the same thing. So, a lot of vectors we'll draw
PxCxlsl_YwY	starting at the origin, but we don't have to.
PxCxlsl_YwY	So, let's just check and see how things went in recitation.
PxCxlsl_YwY	So, let's say that I give you the vector
PxCxlsl_YwY	*amplt;3,2,1amp*gt;. And so, what do you think about
PxCxlsl_YwY	the length of this vector? OK, I see an answer forming.
PxCxlsl_YwY	So, a lot of you are answering the same thing.
PxCxlsl_YwY	Maybe it shouldn't spoil it for those who haven't given it yet.
PxCxlsl_YwY	OK, I think the overwhelming vote is in favor of answer
PxCxlsl_YwY	number two. I see some sixes, I don't know.
PxCxlsl_YwY	That's a perfectly good answer, too, but hopefully in a few
PxCxlsl_YwY	minutes it won't be I don't know anymore.
PxCxlsl_YwY	So, let's see. How do we find -- -- the length
PxCxlsl_YwY	of a vector three, two, one?
PxCxlsl_YwY	Well, so, this vector, A, it comes towards us along
PxCxlsl_YwY	the x axis by three units. It goes to the right along the
PxCxlsl_YwY	y axis by two units, and then it goes up by one unit
PxCxlsl_YwY	along the z axis. OK, so, it's pointing towards
PxCxlsl_YwY	here. That's pretty hard to draw.
PxCxlsl_YwY	So, how do we get its length? Well, maybe we can start with
PxCxlsl_YwY	something easier, the length of the vector in the
PxCxlsl_YwY	plane. So, observe that A is obtained
PxCxlsl_YwY	from a vector, B, in the plane.
PxCxlsl_YwY	Say, B equals three (i) hat plus two (j) hat.
PxCxlsl_YwY	And then, we just have to, still, go up by one unit,
PxCxlsl_YwY	OK? So, let me try to draw a
PxCxlsl_YwY	picture in this vertical plane that contains A and B.
PxCxlsl_YwY	If I draw it in the vertical plane,
PxCxlsl_YwY	so, that's the Z axis, that's not any particular axis,
PxCxlsl_YwY	then my vector B will go here, and my vector A will go above
PxCxlsl_YwY	it. And here, that's one unit.
PxCxlsl_YwY	And, here I have a right angle. So, I can use the Pythagorean
PxCxlsl_YwY	theorem to find that length A^2 equals length B^2 plus one.
PxCxlsl_YwY	Now, we are reduced to finding the length of B.
PxCxlsl_YwY	The length of B, we can again find using the
PxCxlsl_YwY	Pythagorean theorem in the XY plane because here we have the
PxCxlsl_YwY	right angle. Here we have three units,
PxCxlsl_YwY	and here we have two units. OK, so, if you do the
PxCxlsl_YwY	calculations, you will see that,
PxCxlsl_YwY	well, length of B is square root of (3^2 2^2),
PxCxlsl_YwY	that's 13. So, the square root of 13 -- --
PxCxlsl_YwY	and length of A is square root of length B^2 plus one (square
PxCxlsl_YwY	it if you want) which is going to be square root of 13 plus one
PxCxlsl_YwY	is the square root of 14, hence, answer number two which
PxCxlsl_YwY	almost all of you gave. OK, so the general formula,
PxCxlsl_YwY	if you follow it with it, in general if we have a vector
PxCxlsl_YwY	with components a1, a2, a3,
PxCxlsl_YwY	then the length of A is the square root of a1^2 plus a2^2
PxCxlsl_YwY	plus a3^2. OK, any questions about that?
PxCxlsl_YwY	Yes? Yes.
PxCxlsl_YwY	So, in general, we indeed can consider vectors
PxCxlsl_YwY	in abstract spaces that have any number of coordinates.
PxCxlsl_YwY	And that you have more components.
PxCxlsl_YwY	In this class, we'll mostly see vectors with
PxCxlsl_YwY	two or three components because they are easier to draw,
PxCxlsl_YwY	and because a lot of the math that we'll see works exactly the
PxCxlsl_YwY	same way whether you have three variables or a million
PxCxlsl_YwY	variables. If we had a factor with more
PxCxlsl_YwY	components, then we would have a lot of trouble drawing it.
PxCxlsl_YwY	But we could still define its length in the same way,
PxCxlsl_YwY	by summing the squares of the components.
PxCxlsl_YwY	So, I'm sorry to say that here, multi-variable,
PxCxlsl_YwY	multi will mean mostly two or three.
PxCxlsl_YwY	But, be assured that it works just the same way if you have
PxCxlsl_YwY	10,000 variables. Just, calculations are longer.
PxCxlsl_YwY	OK, more questions? So, what else can we do with
PxCxlsl_YwY	vectors? Well, another thing that I'm
PxCxlsl_YwY	sure you know how to do with vectors is to add them to scale
PxCxlsl_YwY	them. So, vector addition,
PxCxlsl_YwY	so, if you have two vectors, A and B, then you can form,
PxCxlsl_YwY	their sum, A plus B. How do we do that?
PxCxlsl_YwY	Well, first, I should tell you,
PxCxlsl_YwY	vectors, they have this double life.
PxCxlsl_YwY	They are, at the same time, geometric objects that we can
PxCxlsl_YwY	draw like this in pictures, and there are also
PxCxlsl_YwY	computational objects that we can represent by numbers.
PxCxlsl_YwY	So, every question about vectors will have two answers,
PxCxlsl_YwY	one geometric, and one numerical.
PxCxlsl_YwY	OK, so let's start with the geometric.
PxCxlsl_YwY	So, let's say that I have two vectors, A and B,
PxCxlsl_YwY	given to me. And, let's say that I thought
PxCxlsl_YwY	of drawing them at the same place to start with.
PxCxlsl_YwY	Well, to take the sum, what I should do is actually
PxCxlsl_YwY	move B so that it starts at the end of A, at the head of A.
PxCxlsl_YwY	OK, so this is, again, vector B. So, observe,
PxCxlsl_YwY	this actually forms, now, a parallelogram,
PxCxlsl_YwY	right? So, this side is,
PxCxlsl_YwY	again, vector A. And now, if we take the
PxCxlsl_YwY	diagonal of that parallelogram, this is what we call A plus B,
PxCxlsl_YwY	OK, so, the idea being that to move along A plus B,
PxCxlsl_YwY	it's the same as to move first along A and then along B,
PxCxlsl_YwY	or, along B, then along A. A plus B equals B plus A.
PxCxlsl_YwY	OK, now, if we do it numerically,
PxCxlsl_YwY	then all you do is you just add the first component of A with

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

a starting point and an ending point.

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

of a vector three, two, one?

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

here. That's pretty hard to draw.

PxCxlsl_YwY

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

PxCxlsl_YwY

something easier, the length of the vector in the

PxCxlsl_YwY

plane. So, observe that A is obtained

PxCxlsl_YwY

from a vector, B, in the plane.

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

OK? So, let me try to draw a

PxCxlsl_YwY

picture in this vertical plane that contains A and B.

PxCxlsl_YwY

If I draw it in the vertical plane,

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

it. And here, that's one unit.

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

The length of B, we can again find using the

PxCxlsl_YwY

Pythagorean theorem in the XY plane because here we have the

PxCxlsl_YwY

right angle. Here we have three units,

PxCxlsl_YwY

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

PxCxlsl_YwY

calculations, you will see that,

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

with components a1, a2, a3,

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

Yes? Yes.

PxCxlsl_YwY

So, in general, we indeed can consider vectors

PxCxlsl_YwY

in abstract spaces that have any number of coordinates.

PxCxlsl_YwY

And that you have more components.

PxCxlsl_YwY

In this class, we'll mostly see vectors with

PxCxlsl_YwY

two or three components because they are easier to draw,

PxCxlsl_YwY

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

PxCxlsl_YwY

same way whether you have three variables or a million

PxCxlsl_YwY

variables. If we had a factor with more

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

by summing the squares of the components.

PxCxlsl_YwY

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

PxCxlsl_YwY

multi will mean mostly two or three.

PxCxlsl_YwY

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

PxCxlsl_YwY

10,000 variables. Just, calculations are longer.

PxCxlsl_YwY

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

PxCxlsl_YwY

vectors? Well, another thing that I'm

PxCxlsl_YwY

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

PxCxlsl_YwY

them. So, vector addition,

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

Well, first, I should tell you,

PxCxlsl_YwY

vectors, they have this double life.

PxCxlsl_YwY

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

PxCxlsl_YwY

draw like this in pictures, and there are also

PxCxlsl_YwY

computational objects that we can represent by numbers.

PxCxlsl_YwY

So, every question about vectors will have two answers,

PxCxlsl_YwY

one geometric, and one numerical.

PxCxlsl_YwY

OK, so let's start with the geometric.

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

of drawing them at the same place to start with.

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

this actually forms, now, a parallelogram,

PxCxlsl_YwY

right? So, this side is,

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

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

PxCxlsl_YwY

OK, now, if we do it numerically,

PxCxlsl_YwY

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