- [Voiceover] So let's say that we've got the curve R

defined, so this is our curve R.

It's X of T times I

plus Y T times J, it's a curve

in two dimensions on the XY plane.

And let's graph it, just graph it in kind of a

a generalized form.

So that's our Y-axis.

This is our X-axis.

Our curve R might look something like this.

It might look something, let me draw a little

bit more of a.

Maybe it looks something like this.

Maybe that's just part of it.

And as T increases, we're going in that direction

right over there.

What I want to do in this video,

this is really more vector algebra than vector calculus.

Is think about at any given point here

whether we can figure out a normal vector.

In particular, a unit normal vector.

Obviously you can figure out a normal vector

you can just divide it by its magnitude

and you will get the unit normal vector.

So I want to figure out at any given point

a vector that's popping straight out in that direction.

And has a magnitude one.

So that would be our unit normal vector.

And to do that, first we'll think about

what a tangent vector is.

And from a tangent vector we can figure out

the normal vector.

And it really goes back to some of what you might

have done in algebra one, or algebra two

of if you have a slope of a line

the negative reciprocal of that slope is going to be

the slope of that negative line.

And we'll see a very similar thing

when we do it right over here with the vector

with this vector algebra.

So the first thing I want to think about is

how do we construct

how do we construct a tangent line.

Well, you could imagine at sum T

at sum T

this is what our position vector is going to look like.

So call that R one.

R one right over there.

And then if we wait, if we allow T to go up a little bit,

if T is time, we'll wait a little while

a few seconds, or however we were measuring things.

And then R two might look something like this.

This is when T has gotten a little bit larger.

We're further down the path.

And so one way that you can approximate

the slope of the tangent line

or the slope between these two points, for now,

is essentially the difference between these two vectors.

The difference between these two vectors is

you could view that

you could view that as

delta

delta R.

This vector plus that vector is equal to that vector.

Or, R two minus R one is going to give you

this delta R right over here.

And as R two, as that increment

between R one and R two gets smaller

and smaller and smaller.

As we have a smaller and smaller T increment

as we get a smaller and smaller T increment.

So we get a smaller and smaller T increment

the slope of that delta R is going to more

and more approximate the slope

of the tangent line.

All the way to the point that if you have

an infinitely small change in T.

So you have a DT.

So you go from R then you just

you change T a very small amount

that delta R, and we can kind of conceptualize that,

as DR, that does approximate

the A tangent vector.

So if you have a very small change in T

then your very small

del DR I'll call it

because now we're talking about a differential.

Your very small differential.

Right over here.

That is a tangent

that is a tangent vector.

So DR

DR is

a tangent

tangent vector at any

at any given point.

And once again, all of this is a little bit of review.

But DR, we can write as

DR is equal to DX

times I plus the

infinite small change in X

times the I unit vector

plus the infinite small change in Y

times the J unit vector.

And you see that, you see that

if I were to draw

if I were to draw a curve.

Let me just draw another one.

Actually, I don't even have to draw the axis.

If our DR looks like that

if that is our DR

then, we can break that down

into its vertical and horizontal components.

This right over here is DY.

And that right over there

that right over there is

that is DX.

And so we see that DX times I.

Actually, this is DX times I.

And this is DY

this is DY times J.

DY is the magnitude,

J gives us the direction.

DX is the magnitude.

I tells us that we're moving

in the horizontal direction.

Over here, this actually would be a negative.

This must be a negative value right over here

and this must be a positive value

based on the way that I drew it.

So that gives us a tangent vector.

And now we want to from that tangent vector

figure out a normal vector.

A vector that is essentially perpendicular

to this vector right over here.

And there's actually going to be two

vectors like that.

There's going to be the vector

that kind of is perpendicular in the right direction

because we care about direction.

Or the vector that's perpendicular in the left direction.

And we can pick either one.

But for this video, I'm gonna focus on the one

that goes in the right direction.

We're gonna see that that's gonna be useful

in the next video when we start doing a little bit of

vector calculus.

And so let's think about what that might be.

And what I'll do to make it a little bit clearer.

Let me draw a DR again.

I'll draw a DR like this.

This is our DR.

This is DR.

And then this, right over here.

This right over there, we already said this is DY

times I.

And then this, sorry,

this is DY times J.

We're going in the vertical direction.

DY times J.

And then in a different color

this right now if I already used that color.

I haven't used, oh,

I haven't used or had blue yet.

So this right over here is DX

DX times I.

So we know from our algebra courses

you take the negative reciprocal

so there's gonna be something about swapping

these two things around, and then taking the

negative one.

But to figure out, we want the one that goes to the right.

So which one should we use?

So let's think about it a little bit.

If we, if we take DY

times I.

So we take this length

in the I direction,

we're gonna get

we're gonna get this.

We're gonna get that, so this is

DY

times I.

And then if we were to

if we were to take D, if we were to just take

DX

times

J, that would take us down.

'Cause DX it must be negative here

since it's pointed to the left.

So we have to swap the sign of DX to go upwards.

So we swap the sign of DX

to go upwards.

'Cause I was here it was a negative sign.

It went leftwards, we want it to go upwards.

So this is gonna be negative

negative DX times J.

We're now moving in the vertical direction.

And that, at least visually, this isn't kind of a

rigorous proof that I'm giving you.

But this is hopefully good a good visual representation

that that does

that that does get you.

I should have drawn it a little bit.

That does get you pretty

that gets you pretty close, just visually inspecting it

to what looks like the perpendicular line.

It's consistent with what you learned in algebra class,

as well.

That we're taking the negative reciprocal,

we're swapping the X's and the Y's.

Or the change in X and the change in Y.

And we're taking the negative of one of them.

And so we have our normal line

just like that.

Our normal vector.

So a normal vector is going to be

DYI

minus DXJ.

But then if we want a normalize it

we want to divide by

by that magnitude.

So a normal, let me write it this way.

A normal vector.

So let me call this

I'll just call it A.

A normal vector

is going to be DY

times I.

Is going to be

DY

times I.

Minus DX times J.

I'll do that same blue color.

Minus DX

times J.

Now, if we want this to be a unit normal vector

we have to divide it by the magnitude of A.

But what is the magnitude of A?

The magnitude of A

is going to be equal to

it's going to be equal to the square root

of, and I'll just start with the DX squared.

So it's the negative DX squared.

Which is just going to be DX squared.

The same thing as positive DX squared.

It's going to be DX squared

plus DY squared.

Plus DY

squared.

I could have put the negative right in here

but then when you square it, that negative would disappear.

But this thing right over here

and we saw this when we first started exploring arc length.

This thing right over here is the exact same thing

as DS,

and I know there's no DS

that we've shown right over here.

But we've seen it multiple times.

When you're thinking of about if you

if you think about the length of DR as DS

that's exactly what this thing over here is.

So this can also be written as

DS.

So the infinite hasn't really changed

in the arc length

but it's a scaler quantity.

You're not concerned

you're just concerned with the absolute distance.

You're not concerned so much with the direction.

Another way to be do it is

it's the magnitude

it's the magnitude right over here of DR.

So now we have everything we need to construct

our unit normal vector.

Our unit normal vector at any point.

And I'll now write N

and I'll put a hat on top of it.

Say that this is a unit normal vector.

We'll have magnitude one is going to be

equal to A divided by this.

Or we could even write it this way.

So we could write it as

there are multiple ways we can write it.

We can write it as

I'll write it in this color.

As DY

times I

minus DX

times J.

And then all of that times

or maybe not times, divided by,

DS.

Divided by the magnitude of this.

So divided by

divided by

DS.

And obviously I can distribute it on each of these

by on each of these terms.

But this right here, we've been able to construct

a unit normal vector at any point

on this curve.