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What I want to do in
this video is show you

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a way to represent a
vector by its components.

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And this is sometimes
called engineering notation

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for vectors.

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But it's super
useful, because it

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allows us to keep track of
the components of the vector

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and it makes it a
little bit tangible

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when we talk about the
individual components.

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So let's break down this
vector right over here.

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I'm just assuming it's
a velocity vector.

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Vector v. Its magnitude
is 10 meters per second.

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And it's pointed in a
direction 30 degrees

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above the horizontal.

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So we've broken down these
vectors in the past before.

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The vertical component
right over here.

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Its magnitude would
be-- so the magnitude

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of the vertical component,
right over here,

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is going to be 10
sine of 30 degrees.

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It's going to be 10 meters per
second times the sine of 30

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degrees.

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This comes from the basic
trigonometry from sohcahtoa.

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And I covered that in more
detail in previous videos.

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Sine of 30 degrees is 1/2.

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So this is going to be 5,
or 5 meters per second.

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10 times 1/2 is 5
meters per second.

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So that's the magnitude
of its vertical component.

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And in the last few
videos, I kind of,

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in a less tangible way of
specifying the vertical vector,

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I often use this notation, which
isn't as tangible as I like.

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And that's why I'm
going to make it

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a little bit better
in this video.

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I said that that vector
itself is 5 meters per second.

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But what I told you is that
the direction is implicitly

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given because this
is a vertical vector.

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And I told you in previous
videos that if it's positive,

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it means up and if it's
negative, it means down.

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So I kind of have to give you
this context here so that you

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can appreciate that this is a
vector that just the sine of it

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is giving you its direction.

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But I have to keep telling
you this is a vertical vector.

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So it wasn't that tangible.

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And so we had the
same issue when

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we talked about the
horizontal vectors.

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So this horizontal
vector right over here,

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the magnitude of this
horizontal vector

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is going to be 10
cosine of 30 degrees.

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And once again, it
comes straight out

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of basic trigonometry.

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10 cosine of 30 degrees.

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And so cosine of 30 degrees is
the square root of 3 over 2.

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Multiply it by 10,
you get 5 square roots

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of 3 meters per second.

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And once again, in
previous videos,

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I used this notation
sometimes, where I was actually

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saying that the vector
is 5 square roots of 3

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meters per second.

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But in order to ensure that
this wasn't just a magnitude,

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I kept having to tell you
in the horizontal direction.

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If it's positive it's
going to the right,

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and if it's negative,
it's going to the left.

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What I want to do
in this video, is

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give us a convention
so that I don't

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have to keep doing
this for the direction.

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And it makes it all a
little bit more tangible.

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And so what we do
is we introduced

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the idea of unit vectors.

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So by definition, we'll
introduce the vector i i.

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Sometimes it's called i hat.

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And I'll draw it like here.

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I'll make it a
little bit smaller.

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So the vector i hat.

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So that right there is a
picture of the vector i hat.

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And we've put a little
hat on top of the i

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to show that it
is a unit vector.

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And what a unit
vector is-- so i hat

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goes in the positive
x direction.

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That's just how it's defined.

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And the unit vector tells
us that its magnitude is 1.

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So the magnitude of the
vector i hat is equal to 1.

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And its direction is in
the positive x direction.

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So if we really
wanted to specify

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this kind of x component
vector in a better way,

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we really should call it
5 square roots of 3 times

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this unit vector.

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Because this green
vector over here

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is going to be 5
square roots of 3 times

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this vector right over here,
because this vector just

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has length 1.

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So it's 5 square roots of
3 times the unit vector.

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And what I like about
this is that now

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I don't have to
tell you, remember,

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this is a horizontal vector.

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Positive is to the right,
negative is to the left.

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It's implicit here.

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Because clearly if this
is a positive value,

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it's going to be a
positive multiple of i.

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It's going to go to the right.

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If it's a negative value,
it flips around the vector

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and then it goes to the left.

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So this is actually a better way
of specifying the x component

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vector.

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Or if I broke it down, this
vector v into its x component,

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this is a better way of
specifying that vector.

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Same thing for the y direction.

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We can define a unit vector.

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And let me pick a color
that I have not used yet.

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Let me find this
pink I haven't used.

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We can find a unit vector
that goes straight up in the y

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direction called unit vector j.

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And once again, the magnitude
of unit vector j is equal to 1.

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This little hat on
top of it tells us--

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or sometimes it's called a
caret character-- that tells us

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that it is a vector,
but it is a unit vector.

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It has a magnitude of 1.

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And by definition, the vector
j goes and has a magnitude of 1

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in the positive y direction.

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So the y component
of this vector,

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instead of saying it's
5 meters per second

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in the upwards direction or
instead of saying that it's

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implicitly upwards because
it's a vertical vector

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or it's a vertical
component and it's positive,

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we can now be a little bit
more specific about it.

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We can say that it is
equal to 5 times j.

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Because you see, this
magenta vector, it's

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going the exact same direction
as j, it's just 5 times longer.

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I don't know if it's
exactly 5 times.

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I'm just trying to
estimate it right now.

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It's just 5 times longer.

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Now what's really
cool about this

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is besides just being able to
express the components as now

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multiples of explicit
vectors, instead of just being

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able to do that--
which we did do,

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we're representing
the components

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as explicit vectors--
we also know

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that the vector v is the
sum of its components.

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If you start with this
green vector right here

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and you add this vertical
component right over here,

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you have head to tails.

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You get the blue vector.

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And so we can actually
use the components

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to represent the vector itself.

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We don't always have
to draw it like this.

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So we can write that
vector v is equal to-- let

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me write it this way-- it's
equal to its x component

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vector plus the y
component vector.

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And we can write
that, the x component

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vector is 5 square
roots of 3 times i.

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And then it's going to
be plus the y component,

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the vertical component, which
is 5j, which is 5 times j.

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And so what's really neat here
is now you could specify any

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vector in two dimensions
by some combination

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of i's and j's or some scaled
up combinations of i's and j's.

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And if you want to go
into three dimensions,

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and you often will, especially
as the physics class

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moves on through
the year, you can

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introduce a vector in
the positive z direction,

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depending on how
you want to do it.

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Although z is
normally up and down.

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But whatever the
next dimension is,

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you can define a vector k that
goes into that third dimension.

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Here I'll do it in a kind
of unconventional way.

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I'll make k go in
that direction.

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Although the standard
convention when you do it

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in three dimensions is that k
is the up and down dimension.

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But this by itself is
already pretty neat

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because we can now represent any
vector through its components

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and it's also going to
make the math much easier.