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# Unit vector notation

Expressing a vector as the scaled sum of unit vectors. Created by Sal Khan.

Video transcript

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 n-dimensional vectors—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.