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- [Voiceover] All the problems

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we've been dealing with so far

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have essentially been
happening in one dimension.

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You could go forward or back.

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So you could go forward or back.

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Or right or left.

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Or you could go up or down.

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What I wanna start to
talk about in this video

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is what happens when we
extend that to two dimensions

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or we can even just extend
what we're doing in this video

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to three or four, really an
arbitrary number of dimensions.

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Although if you're dealing
with classical mechanics

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you normally don't have to go
more than three dimensions.

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And if you're gonna deal
with more than one dimension,

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especially in two dimensions,

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we're also gonna be dealing
with two-dimensional vectors.

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And I just wanna make
sure, through this video,

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that we understand at least the basics

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of two-dimensional vectors.

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Remember, a vector is something

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that has both magnitude and direction.

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So the first thing I wanna do

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is just give you a visual understanding

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of how vectors in two
dimensions would add.

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So let's say I have a vector right here.

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That is vector A.

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So, once again, its magnitude is specified

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by the length of this arrow.

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And its direction is specified

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by the direction of the arrow.

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So it's going in that direction.

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Now let's say I have another vector.

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Let's call it vector B.

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Let's call it vector B.

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It looks like this.

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Now what I wanna do in this video

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is think about what happens when I add

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vector A to vector B.

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So there's a couple things to think about

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when you visually depict vectors.

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The important thing is,
for example, for vector A,

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that you get the length right

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and you get the direction right.

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Where you actually draw it doesn't matter.

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So this could be vector A.

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This could also be vector A.

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Notice, it has the same length

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and it has the same direction.

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This is also vector A.

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I could draw vector A up
here. It does not matter.

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I could draw vector A up there.

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I could draw vector B.

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I could draw vector B over here.

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It's still vector B.

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It still has the same
magnitude and direction.

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Notice, we're not saying

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that its tail has to
start at the same place

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that vector A's tail starts at.

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I could draw vector B over here.

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So I can always have the same vector

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but I can shift it around.

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So I can move it up there.

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As long as it has the same
magnitude, the same length,

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and the same direction.

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And the whole reason I'm doing that

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is because the way to
visually add vectors...

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If I wanted to add vector A

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plus vector B...

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And I'll show you how to
do it more analytically

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in a future video.

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I can literally draw vector A.

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I draw vector A.

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So that's vector A, right over there.

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And then I can draw vector B,

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but I put the tail of vector
B to the head of vector A.

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So I shift vector B over

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so its tail is right at
the head of vector A.

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And then vector B would
look something like this.

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It would look something like this.

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And then if you go from the tail of A

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all the way to the head of B,

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all the way to the head of B,

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and you call that vector C,

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that is the sum of A and B.

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And it should make sense,
if you think about it.

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Let's say these were displacement vectors.

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So A shows that you're being displaced

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this much in this direction.

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B shows that you're being displaced

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this much in this direction.

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So the length of B in that direction.

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And if I were to say you
have a displacement of A,

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and then you have a displacement of B,

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what is your total displacement?

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So you would have had to be,

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I guess, shifted this
far in this direction,

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and then you would be shifted
this far in this direction.

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So the net amount that you've been shifted

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is this far in that direction.

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So that's why this would
be the sum of those.

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Now we can use that same idea

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to break down any vector in two dimensions

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into, we could say, into its components.

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And I'll give you a better sense

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of what that means in a second.

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So if I have vector A.

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Let me pick a new letter.

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Let's call this vector "vector X."

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Let's call this "vector X."

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I can say that vector X
is going to be the sum of

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this vector right here in green

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and this vector right here in red.

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Notice, X starts at the
tail of the green vector

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and goes all the way to the
head of the magenta vector.

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And the magenta vector starts

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at the head of the green vector

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and then finishes, I guess,

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well where it finishes is
where vector X finishes.

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And the reason why I do this...

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And, you know, hopefully from this

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comparable explanation right here,

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says, okay, look, the green
vector plus the magenta vector

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gives us this X vector.

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That should make sense.

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I put the head of the green vector

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to the tail of this magenta
vector right over here.

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But the whole reason why I did this is,

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if I can express X as a
sum of these two vectors,

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it then breaks down X into
its vertical component

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and its horizontal component.

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So I could call this

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the horizontal component,

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or I should say the vertical component.

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X vertical.

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And then I could call this over here

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the X horizontal.

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Or another way I could draw it,

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I could shift this X vertical over.

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Remember, it doesn't
matter where I draw it,

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as long as it has the same
magnitude and direction.

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And I could draw it like this.

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X vertical.

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And so what you see is

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is that you could express this vector X...

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Let me do it in the same colors.

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You can express this vector X

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as the sum of its horizontal
and its vertical components.

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As the sum of its horizontal
and its vertical components.

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Now we're gonna see over and over again

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that this is super powerful

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because what it can do is

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it can turn a two-dimensional problem

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into two separate
one-dimensional problems,

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one acting in a horizontal direction,

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one acting in a vertical direction.

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Now let's do it a little
bit more mathematical.

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I've just been telling you
about length and all of that.

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But let's actually break down...

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Let me just show you what this means,

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to break down the components of a vector.

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So let's say that I have a
vector that looks like this.

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Let me do my best to...

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Let's say I have a vector
that looks like this.

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It's length is five.

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So let me call this vector A.

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So vector A's length is equal to five.

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And let's say that its direction...

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We're gonna give its
direction by the angle

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between the direction its pointing in

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and the positive X axis.

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So maybe I'll draw an axis over here.

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So let's say that this right over here

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is the positive Y axis going
in the vertical direction.

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This right over here
is the positive X axis

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going in the horizontal direction.

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And to specify this vector's direction

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I will give this angle right over here.

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And I'm gonna give a very peculiar angle,

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but I picked this for a specific reason,

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just so things work out neatly in the end.

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And I'm gonna give it in degrees.

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It's 36.8699 degrees.

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So I'm picking that particular number

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for a particular reason.

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Now what I wanna do is I wanna figure out

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this vector's horizontal
and vertical component.

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So I wanna break it down

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into something that's
going straight up or down

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and something that's going
straight right or left.

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So how do I do this?

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Well, one, I could just
draw them, visually,

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see what they look like.

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So its vertical component
would look like this.

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It would start...

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Its vertical component
would look like this.

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And its horizontal component
would look like this.

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Its horizontal component
would look like this.

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The horizontal component,
the way I drew it,

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it would start where vector A starts

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and go as far in the X
direction as vector A's tip,

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but only in the X direction,

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and then you need to, to get
back to the head of vector A,

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you need to have its vertical component.

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And we can sometimes call this,

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we could call the vertical
component over here A sub Y,

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just so that it's moving
in the Y direction.

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And we can call this
horizontal component A sub X.

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Now what I wanna do is I wanna figure out

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the magnitude of A sub Y and A sub X.

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So how do we do that?

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Well, the way we drew this,

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I've essentially set up
a right triangle for us.

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This is a right triangle.

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We know the length of this triangle,

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or the length of this side, or
the length of the hypotenuse.

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That's going to be the
magnitude of vector A.

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And so the magnitude of
vector A is equal to five.

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We already knew that up here.

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So how do we figure out the sides?

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Well, we could use a little
bit of basic trigonometry.

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If we know the angle, and
we know the hypotenuse,

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how do we figure out the
opposite side to the angle?

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So this right here,

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this right here is the
opposite side to the angle.

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And if we forgot some of
our basic trigonometry

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we can relearn it right now.

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Soh-cah-toa.

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Sine is opposite over hypotenuse.

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Cosine is adjacent over hypotenuse.

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Tangent is opposite over adjacent.

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So we have the angle,
we want the opposite,

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and we have the hypotenuse.

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So we could say

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that the sine of our angle,

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the sine of

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36.899 degrees,

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is going to be equal to the
opposite over the hypotenuse.

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The opposite side of the angle

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is the magnitude of our Y component.

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...is going to be equal to the
magnitude of our Y component,

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the magnitude of our Y component,

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over the magnitude of the hypotenuse,

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over this length over here,

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which we know is going
to be equal to five.

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Or if you multiply both sides by five,

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you get five sine

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of 36.899 degrees,

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is equal to the magnitude
of the vertical component

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of our vector A.

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Now before I take out the calculator

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and figure out what this is,

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let me do the same thing for
the horizontal component.

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Over here we know this side
is adjacent to the angle.

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And we know the hypotenuse.

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And so cosine deals with
adjacent and hypotenuse.

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So we know that the
cosine of 36.899 degrees

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is equal to...

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Cosine is adjacent over hypotenuse.

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So it's equal to the
magnitude of our X component

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over the hypotenuse.

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00:10:39,808 --> 00:10:41,410
The hypotenuse here has...

262
00:10:41,410 --> 00:10:43,686
Or the magnitude of the
hypotenuse, I should say,

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00:10:43,686 --> 00:10:45,381
which has a length of five.

264
00:10:45,381 --> 00:10:47,656
Once again, we multiply
both sides by five,

265
00:10:47,656 --> 00:10:51,823
and we get five times the
cosine of 36.899 degrees

266
00:10:53,136 --> 00:10:56,886
is equal to the magnitude
of our X component.

267
00:10:58,500 --> 00:11:00,288
So let's figure out what these are.

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00:11:00,288 --> 00:11:02,099
Let me get the calculator out.

269
00:11:02,099 --> 00:11:04,682
Let me get my trusty TI-85 out.

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00:11:05,582 --> 00:11:08,020
I wanna make sure it's in degree mode.

271
00:11:08,020 --> 00:11:09,553
So let me check.

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00:11:09,553 --> 00:11:11,782
Yep, we're in degree
mode right over there.

273
00:11:11,782 --> 00:11:14,197
Don't wanna... Make sure
we're not in radian mode.

274
00:11:14,197 --> 00:11:15,834
Now let's exit that.

275
00:11:15,834 --> 00:11:20,001
And we have the vertical
component is equal to five times

276
00:11:20,969 --> 00:11:23,219
the sine of 36.899 degrees,

277
00:11:27,679 --> 00:11:31,744
which is, if we round
it, right at about three.

278
00:11:31,744 --> 00:11:33,625
So this is equal to...

279
00:11:33,625 --> 00:11:36,179
So the magnitude of our vertical component

280
00:11:36,179 --> 00:11:37,679
is equal to three.

281
00:11:39,964 --> 00:11:41,218
And then let's do the same thing

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00:11:41,218 --> 00:11:43,635
for our horizontal component.

283
00:11:45,865 --> 00:11:47,948
So now we have five times

284
00:11:49,734 --> 00:11:52,151
the cosine of 36.899 degrees,

285
00:11:54,489 --> 00:11:57,780
is, if once again we round it to, I guess,

286
00:11:57,780 --> 00:12:01,375
our hundredths place,
we get it to being four.

287
00:12:01,375 --> 00:12:03,625
So we get it to being four.

288
00:12:04,718 --> 00:12:07,737
So we see here is a
situation where we have...

289
00:12:07,737 --> 00:12:11,243
This is a classic three-four-five
Pythagorean triangle.

290
00:12:11,243 --> 00:12:15,410
The magnitude of our
horizontal component is four.

291
00:12:17,130 --> 00:12:21,270
The magnitude of our vertical
component, right over here,

292
00:12:21,270 --> 00:12:22,601
is equal to three.

293
00:12:22,601 --> 00:12:23,855
And once again, you might say,

294
00:12:23,855 --> 00:12:27,129
Sal, why are we going
through all of this trouble?

295
00:12:27,129 --> 00:12:27,962
And we'll see in the next video

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00:12:27,962 --> 00:12:30,133
that if we say something has a velocity,

297
00:12:30,133 --> 00:12:32,641
in this direction, of
five meters per second,

298
00:12:32,641 --> 00:12:36,426
we could break that down into
two component velocities.

299
00:12:36,426 --> 00:12:39,096
We could say that that's
going in the upwards direction

300
00:12:39,096 --> 00:12:40,512
at three meters per second,

301
00:12:40,512 --> 00:12:43,601
and it's also going to the right
in the horizontal direction

302
00:12:43,601 --> 00:12:45,342
at four meters per second.

303
00:12:45,342 --> 00:12:47,107
And it allows us to break up the problem

304
00:12:47,107 --> 00:12:48,523
into two simpler problems,

305
00:12:48,523 --> 00:12:50,334
into two one-dimensional problems,

306
00:12:50,334 --> 00:00:00,000
instead of a bigger two-dimensional one.