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Good afternoon.

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We've done a lot of
work with vectors.

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In a lot of the problems, when
we launch something into--- In

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the projectile motion problems,
or when you were

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doing the incline plane
problems. I always gave you a

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vector, like I would draw
a vector like this.

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I would say something
has a velocity of

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10 meters per second.

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It's at a 30 degree angle.

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And then I would break it up
into the x and y components.

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So if I called this vector v, I
would use a notation, v sub

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x, and the v sub x would have
been this vector right here.

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v sub x would've been this
vector down here.

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The x component of the vector.

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And then v sub y would have been
the y component of the

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vector, and it would have
been this vector.

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So this was v sub x,
this was v sub y.

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And hopefully by now, it's
second nature of how we would

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figure these things out. v sub
x would be 10 times cosine of

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this angle.

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10 cosine of 30 degrees, which I
think is square root of 3/2,

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but we're not worried about
that right now.

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And v sub y would be 10 times
the sine of that angle.

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This hopefully should be
second nature to you.

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If it's not, you can just go
through SOH-CAH-TOA and say,

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well, the sine of 30 degrees
is the opposite of the

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hypotenuse.

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And you would get
back to this.

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But we've reviewed all of that,
and you should review

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the initial vector videos.

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But what I want you to do now,
because this is useful for

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simple projectile motion
problems-- But once we start

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dealing with more complicated
vectors-- and maybe we're

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dealing with multi-dimensional
of vectors, three-dimensional

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vectors, or we start doing
linear algebra, where we do

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end dimensional factors --we
need a coherent way, an

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analytical way, instead of
having to always draw a

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picture of representing
vectors.

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So what we do is, we use
something I call, and I think

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everyone calls it, unit
vector notation.

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So what does that mean?

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So we define these
unit vectors.

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Let me draw some axes.

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And it's important to keep in
mind, this might seem a little

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confusing at first, but this
is no different than what

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we've been doing in our physics
problem so far.

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Let me draw the axes
right there.

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Let's say that this is 1,
this is 0, this is 2.

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0, 1, 2.

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I don't know if must been
writing an Arabic or

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something, going backwards.

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This is 0, 1, 2,
that's not 20.

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And then let's say this is 1,
this is 2, in the y direction.

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I'm going to define what I call
the unit vectors in two

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dimensions.

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So I'm going to first
define a vector.

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I'll call this vector i.

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And this is the vector.

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It just goes straight in the x
direction, has no y component,

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and it has the magnitude of 1.

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And so this is i.

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We denote the unit vector
by putting this little

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cap on top of it.

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There's multiple notations.

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Sometimes in the book, you'll
see this i without the cap,

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and it's just boldface.

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There's some other notations.

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But if you see i, and not in
the imaginary number sense,

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you should realize that that's
the unit vector.

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It has magnitude 1 and it's
completely in the x direction.

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And I'm going to define another
vector, and that one

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is called j.

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And that is the same thing
but in the y direction.

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That is the vector j.

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You put a little cap over it.

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So why did I do this?

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Well, if I'm dealing with
two dimensions.

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And as later we'll see in three
dimensions, so there

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will actually be a third
dimension and we'll call that

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k, but don't worry about
that right now.

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But if we're dealing in two
dimensions, we can define any

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vector in terms of some sum
of these two vectors.

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So how does that work?

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Well, this vector here,
let's call it v.

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This vector, v, is
the sum of its x

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component plus its y component.

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When you add vectors, you
can put them head

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to tail like this.

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And that's the sum.

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So hopefully knowing what we
already know, we knew that the

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vector, v, is equal to its x

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component plus its y component.

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When you add vectors, you
essentially just put

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them head to tails.

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And then the resulting sum
is where you end up.

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It would be if you added this
vector, and then you put this

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tail to this head.

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And you end up there.

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So you end up there.

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So that's the vector.

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So can we define v sub x as some
multiple of i, of this

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unit vector?

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Well, sure.

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v sub x completely goes
in the x direction.

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But it doesn't have
a magnitude of 1.

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It has a magnitude of 10
cosine 30 degrees.

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So its magnitude is ten.

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Let me draw the unit
vector up here.

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This is the unit vector i.

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It's going to look something
like this and this.

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So v sub x is in the exact same
direction, and it's just

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a scaled version of
this unit vector.

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And what multiple is it
of that unit vector?

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Well, the unit vector has
a magnitude of 1.

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This has a magnitude of 10
cosine of 30 degrees.

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I think that's like, 5
square roots of 3, or

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something like that.

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So we can write v sub x-- I keep
switching colors to keep

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things interesting.

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We can write v sub x is equal
to 10 cosine of 30 degrees

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times-- that's the degrees
--times the unit vector i--

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let me stay in that color, so
you don't confused --times the

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unit vector i.

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Does that make sense?

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Well, the unit vector i goes in
the exact same direction.

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But the x component of this
vector is just a lot longer.

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It's 10 cosine 30
degrees long.

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And that's equal to-- cosine of
30 degrees is square root

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of 3/2 --so that's 5 square
roots of 3 i.

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Similary, we can write the y
component of this vector as

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some multiple of j.

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So we could say v sub y, the y
component-- Well, what is sine

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of 30 degrees?

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Sine of 30 degrees is 1/2.

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1/2 times 10, so this is 5.

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So the y component goes
completely in the y direction.

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So it's just going to be a
multiple of this vector j, of

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the unit vector j.

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And what multiple is it?

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Well, it has length 5,
while the unit vector

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has just length 1.

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So it's just 5 times
the unit vector j.

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So how can we write vector v?

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Well, we know the vector v is
the sum of its x component and

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its y component.

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And we also know, so this
is a whole vector v.

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What's its x component?

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Its x component can be
written as a multiple

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of the x unit vector.

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That's that right there.

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So you can write it as
5 square roots of 3

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i plus its y component.

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So what's its y component?

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Well, its y component is just
a multiple of the y unit

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vector, which is called
j, with the little

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funny hat on top.

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And that's just this.

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It's 5 times j.

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So what we've done now, by
defining these unit vectors--

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And I can switch this
color just so you

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remember this is i.

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This unit vector is this.

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Using unit vectors in two
dimensions, and we can

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eventually do them in multiple
dimensions, we can

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analytically express any
two dimensional vector.

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Instead of having to always draw
it like we did before,

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and having to break out
its components and

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always do it visually.

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We can stay in analytical mode
and non graphical mode.

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And what makes this very useful
is that if I can write

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a vector in this format, I can
add them and subtract them

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without having to resort
to visual means.

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And what do I mean by that?

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So if I had to find some vector
a, is equal to, I don't

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know, 2i plus 3j.

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And I have some other
vector b.

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This little arrow just
means it's a vector.

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Sometimes you'll see it
as a whole arrow.

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As, I don't know, 10i plus 2j.

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If I were to say what's
the sum of these two

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vectors a plus b?

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Before we had this unit vector
notation, we would have to

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draw them, and put them
heads to tails.

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And you had to do it visually,
and it would take

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you a lot of time.

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But once you have it broken up
into the x and y components,

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you can just separately add
the x and y components.

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So vector a plus vector b,
that's just 2 plus 10 times i

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plus 3 plus 2 times j.

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And that's equal
to 12i plus 5j.

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And something you might want to
do, maybe I'll do it in the

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future video, is actually draw
out these two vectors and add

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them visually.

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And you'll see that you
get this exact answer.

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And as we go into further
videos, or future videos,

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you'll see how this is super
useful once we start doing

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more complicated physics
problems, or once we start

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doing physics with calculus.

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Anyway, I'm about to run out
of time on the ten minutes.

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So I'll see you in
the next video.

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