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Constructing a unit normal vector to a curve

Figuring out a unit normal vector at any point along a curve defined by a position vector function. Created by Sal Khan.

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Video transcript

- [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. 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.