Split (1)

train · 3.69k rows

video_id stringclasses 7 values	text stringlengths 2 29.3k
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,

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.