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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
***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