Good afternoon.
We've done a lot of work with vectors.
In a lot of the problems, when we launch something into--- In
the projectile motion problems, or when you were
doing the incline plane problems. I always gave you a
vector, like I would draw a vector like this.
I would say something has a velocity of
10 meters per second.
It's at a 30 degree angle.
And then I would break it up into the x and y components.
So if I called this vector v, I would use a notation, v sub
x, and the v sub x would have been this vector right here.
v sub x would've been this vector down here.
The x component of the vector.
And then v sub y would have been the y component of the
vector, and it would have been this vector.
So this was v sub x, this was v sub y.
And hopefully by now, it's second nature of how we would
figure these things out. v sub x would be 10 times cosine of
this angle.
10 cosine of 30 degrees, which I think is square root of 3/2,
but we're not worried about that right now.
And v sub y would be 10 times the sine of that angle.
This hopefully should be second nature to you.
If it's not, you can just go through SOH-CAH-TOA and say,
well, the sine of 30 degrees is the opposite of the
hypotenuse.
And you would get back to this.
But we've reviewed all of that, and you should review
the initial vector videos.
But what I want you to do now, because this is useful for
simple projectile motion problems-- But once we start
dealing with more complicated vectors-- and maybe we're
dealing with multi-dimensional of vectors, three-dimensional
vectors, or we start doing linear algebra, where we do
end dimensional factors --we need a coherent way, an
analytical way, instead of having to always draw a
picture of representing vectors.
So what we do is, we use something I call, and I think
everyone calls it, unit vector notation.
So what does that mean?
So we define these unit vectors.
Let me draw some axes.
And it's important to keep in mind, this might seem a little
confusing at first, but this is no different than what
we've been doing in our physics problem so far.
Let me draw the axes right there.
Let's say that this is 1, this is 0, this is 2.
0, 1, 2.
I don't know if must been writing an Arabic or
something, going backwards.
This is 0, 1, 2, that's not 20.
And then let's say this is 1, this is 2, in the y direction.
I'm going to define what I call the unit vectors in two
dimensions.
So I'm going to first define a vector.
I'll call this vector i.
And this is the vector.
It just goes straight in the x direction, has no y component,
and it has the magnitude of 1.
And so this is i.
We denote the unit vector by putting this little
cap on top of it.
There's multiple notations.
Sometimes in the book, you'll see this i without the cap,
and it's just boldface.
There's some other notations.
But if you see i, and not in the imaginary number sense,
you should realize that that's the unit vector.
It has magnitude 1 and it's completely in the x direction.
And I'm going to define another vector, and that one
is called j.
And that is the same thing but in the y direction.
That is the vector j.
You put a little cap over it.
So why did I do this?
Well, if I'm dealing with two dimensions.
And as later we'll see in three dimensions, so there
will actually be a third dimension and we'll call that
k, but don't worry about that right now.
But if we're dealing in two dimensions, we can define any
vector in terms of some sum of these two vectors.
So how does that work?
Well, this vector here, let's call it v.
This vector, v, is the sum of its x
component plus its y component.
When you add vectors, you can put them head
to tail like this.
And that's the sum.
So hopefully knowing what we already know, we knew that the
vector, v, is equal to its x
component plus its y component.
When you add vectors, you essentially just put
them head to tails.
And then the resulting sum is where you end up.
It would be if you added this vector, and then you put this
tail to this head.
And you end up there.
So you end up there.
So that's the vector.
So can we define v sub x as some multiple of i, of this
unit vector?
Well, sure.
v sub x completely goes in the x direction.
But it doesn't have a magnitude of 1.
It has a magnitude of 10 cosine 30 degrees.
So its magnitude is ten.
Let me draw the unit vector up here.
This is the unit vector i.
It's going to look something like this and this.
So v sub x is in the exact same direction, and it's just
a scaled version of this unit vector.
And what multiple is it of that unit vector?
Well, the unit vector has a magnitude of 1.
This has a magnitude of 10 cosine of 30 degrees.
I think that's like, 5 square roots of 3, or
something like that.
So we can write v sub x-- I keep switching colors to keep
things interesting.
We can write v sub x is equal to 10 cosine of 30 degrees
times-- that's the degrees --times the unit vector i--
let me stay in that color, so you don't confused --times the
unit vector i.
Does that make sense?
Well, the unit vector i goes in the exact same direction.
But the x component of this vector is just a lot longer.
It's 10 cosine 30 degrees long.
And that's equal to-- cosine of 30 degrees is square root
of 3/2 --so that's 5 square roots of 3 i.
Similary, we can write the y component of this vector as
some multiple of j.
So we could say v sub y, the y component-- Well, what is sine
of 30 degrees?
Sine of 30 degrees is 1/2.
1/2 times 10, so this is 5.
So the y component goes completely in the y direction.
So it's just going to be a multiple of this vector j, of
the unit vector j.
And what multiple is it?
Well, it has length 5, while the unit vector
has just length 1.
So it's just 5 times the unit vector j.
So how can we write vector v?
Well, we know the vector v is the sum of its x component and
its y component.
And we also know, so this is a whole vector v.
What's its x component?
Its x component can be written as a multiple
of the x unit vector.
That's that right there.
So you can write it as 5 square roots of 3
i plus its y component.
So what's its y component?
Well, its y component is just a multiple of the y unit
vector, which is called j, with the little
funny hat on top.
And that's just this.
It's 5 times j.
So what we've done now, by defining these unit vectors--
And I can switch this color just so you
remember this is i.
This unit vector is this.
Using unit vectors in two dimensions, and we can
eventually do them in multiple dimensions, we can
analytically express any two dimensional vector.
Instead of having to always draw it like we did before,
and having to break out its components and
always do it visually.
We can stay in analytical mode and non graphical mode.
And what makes this very useful is that if I can write
a vector in this format, I can add them and subtract them
without having to resort to visual means.
And what do I mean by that?
So if I had to find some vector a, is equal to, I don't
know, 2i plus 3j.
And I have some other vector b.
This little arrow just means it's a vector.
Sometimes you'll see it as a whole arrow.
As, I don't know, 10i plus 2j.
If I were to say what's the sum of these two
vectors a plus b?
Before we had this unit vector notation, we would have to
draw them, and put them heads to tails.
And you had to do it visually, and it would take
you a lot of time.
But once you have it broken up into the x and y components,
you can just separately add the x and y components.
So vector a plus vector b, that's just 2 plus 10 times i
plus 3 plus 2 times j.
And that's equal to 12i plus 5j.
And something you might want to do, maybe I'll do it in the
future video, is actually draw out these two vectors and add
them visually.
And you'll see that you get this exact answer.
And as we go into further videos, or future videos,
you'll see how this is super useful once we start doing
more complicated physics problems, or once we start
doing physics with calculus.
Anyway, I'm about to run out of time on the ten minutes.
So I'll see you in the next video.