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## Unit vectors

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# Unit vectors intro

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## Video transcript

We've already seen
that you can visually represent a vector as an arrow,
where the length of the arrow is the magnitude of the vector
and the direction of the arrow is the direction of the vector. And if we want to represent
this mathematically, we could just think
about, well, starting from the tail of
the vector, how far away is the head of the vector
in the horizontal direction? And how far away is it in
the vertical direction? So for example, in the
horizontal direction, you would have to
go this distance. And then in the
vertical direction, you would have to
go this distance. Let me do that in
a different color. You would have to go this
distance right over here. And so let's just say
that this distance is 2 and that this distance is 3. We could represent
this vector-- and let's call this vector v. We
could represent vector v as an ordered list or a
2-tuple of-- so we could say we move 2 in the
horizontal direction and 3 in the vertical direction. So you could represent
it like that. You could represent
vector v like this, where it is 2
comma 3, like that. And what I now want
to introduce you to-- and we could come up with
other ways of representing this 2-tuple-- is
another notation. And this really
comes out of the idea of what it means to
add and scale vectors. And to do that,
we're going to define what we call unit vectors. And if we're in
two dimensions, we define a unit vector for
each of the dimensions we're operating in. If we're in three dimensions,
we would define a unit vector for each of the three dimensions
that we're operating in. And so let's do that. So let's define a unit vector i. And the way that we denote
that is the unit vector is, instead of putting
an arrow on top, we put this hat on top of it. So the unit vector
i, if we wanted to write it in this
notation right over here, we would say it only goes 1 unit
in the horizontal direction, and it doesn't go at all
in the vertical direction. So it would look
something like this. That is the unit vector i. And then we can define
another unit vector. And let's call
that unit vector-- or it's typically
called j, which would go only in the
vertical direction and not in the horizontal direction. And not in the
horizontal direction, and it goes 1 unit in
the vertical direction. So this went 1 unit
in the horizontal. And now j is going to go
1 unit in the vertical. So j-- just like that. Now any vector, any
two dimensional vector, we can now represent as a sum of
scaled up versions of i and j. And you say, well,
how do we do that? Well, you could imagine
vector v right here is the sum of a vector
that moves purely in the horizontal direction
that has a length 2, and a vector that moves purely
in the vertical direction that has length 3. So we could say
that vector v-- let me do it in that
same blue color-- is equal to-- so if we want
a vector that has length 2 and it moves purely in
the horizontal direction, well, we could just scale
up the unit vector i. We could just
multiply 2 times i. So let's do that-- is equal
to 2 times our unit vector i. So 2i is going to
be this whole thing right over here or
this whole vector. Let me do it in
this yellow color. This vector right over
here, you could view as 2i. And then to that, we're going to
add 3 times j-- so plus 3 times j. Let me write it like this. Let me get that color. Once again, 3 times
j is going to be this vector right over here. And if you add this yellow
vector right over here to the magenta vector,
you're going to get-- notice, we're putting the tail
of the magenta vector at the head of
the yellow vector. And if you start at the
tail of the yellow vector and you go all the way to the
head of the magenta vector, you have now constructed
vector v. So vector v, you could represent it as a
column vector like this, 2 3. You could represent
it as 2 comma 3, or you could represent it as
2 times i with this little hat over it, plus 3 times j,
with this little hat over it. i is the unit vector in
the horizontal direction, in the positive
horizontal direction. If you want to go
the other way, you would multiply it by a negative. And j is the unit vector
in the vertical direction. As we'll see in future
videos, once you go to three dimensions,
you'll introduce a k. But it's very natural to
translate between these two things. Notice, 2, 3-- 2, 3. And so with that, let's actually
do some vector operations using this notation. So let's say that I
define another vector. Let's say it is vector b. I'll just come up with
some numbers here. Vector b is equal to negative 1
times i-- times the unit vector i-- plus 4 times the unit vector
in the horizontal direction. So given these two
vector definitions, what would the would be the
vector v plus b be equal to? And I encourage you to pause
the video and think about it. Well once again,
we just literally have to add
corresponding components. We could say, OK,
well let's think about what we're doing in
the horizontal direction. We're going 2 in the
horizontal direction here, and now we're going negative 1. So our horizontal
component is going to be 2 plus negative
1-- 2 plus negative 1 in the horizontal direction. And we're going to multiply
that times the unit vector i. And this, once
again, just goes back to adding the corresponding
components of the vector. And then we're going to have
plus 4, or plus 3 plus 4-- And let me write it that way--
times the unit vector j in the vertical direction. And so that's going
to give us-- I'll do this all in this one color--
2 plus negative 1 is 1i. And we could literally
write that just as i. Actually, let's do that. Let's just write that as i. But we got that from 2
plus negative 1 is 1. 1 times the vector is just
going to be that vector, plus 3 plus 4 is 7-- 7j. And you see, this is exactly
how we saw vector addition in the past, is that we
could also represent vector b like this. We could represent it
like this-- negative 1, 4. And so if you were
to add v to b, you add the corresponding terms. So if we were to add
corresponding terms, looking at them as column
vectors, that is going to be equal to 2 plus
negative 1, which is 1. 3 plus 4 is 7. So this is the exact same
representation as this. This is using unit
vector notation, and this is representing
it as a column vector.