Differentiating parametric curves
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Vector-valued functions intro
- [Voiceover] Let's say I have some curve C, and it's described, it can be parameterized, I can't say that word, as, let's say x is equal to x of t, y is equal to some function y of t, and let's say that this is valid for t is between a and b. So, t is greater than or equal to a and then less than or equal to b. So, if I were to just draw this on, let me see, I could draw it like this. I'm staying very abstract right now. This is not a very specific example. This is the x axis, this is the y axis. My curve, let's say this is when t is equal to a and then the curve might do something like this. I don't know what it does, let's say it's over there. This is t is equal to b. This actual point right here will be x of b. That would be the x coordinate, you evaluate this function at b, and y of b. And this is, of course, when t is equal to a. The actual coordinate in R2 on the Cartesian coordinates will be x of a, which is this right here, and then y of a, which is that right there. And we've seen that before, that's just a standard way of describing a parametric equation or curve using two parametric equations. What I want to do now is describe this same exact curve using a vector valued function. So, if I define a vector valued function, and if you don't remember what those are we'll have a little bit of review here. Let me say I have a vector valued function r, and I'll put a little vector arrow on top of it. In a lot of textbooks, they'll just bold it and they'll leave scalar valued functions unbolded, but it's hard to draw bold so I'll put a little vector on top. And let's say that r is a function of t. And these are going to be position vectors. Position vectors. Position vectors. And I'm specifying that because in general, when someone talks about a vector, this vector and this vector are considered equivalent as long as they have the same magnitude and direction. No one really cares about what their start and end points are as long as their direction is the same and their length is the same. But, when you talk about position vectors, you're saying, "No, these vectors are all going to start at zero, at the origin." And when you state the position vector you're implicitly saying, "This is specifying a unique position." In this case, it's going to be in two dimensional space, but it could be in three dimensional space, or really four, five, whatever, n dimensional space. So, when you state the position vector, you're literally saying, "Okay, this vector literally specifies that point in space." So, let's see if we can describe this curve as a vector, a position vector valued function. So, we can say r of t, let me switch back to that pink color, oh, this keeps staying green, is equal to x of t times the unit vector in the x direction, the unit vector gets a little caret on top, a little hat that's like the arrow for it. That just says it's a unit vector. Plus, y of t times j. If I was dealing with a curve in three dimensions, I would have, plus z of t times k, but we're dealing with two dimensions right here. And so, the way this works is, you're just taking your ... for any t, and still we're gonna have t is greater than or equal to a and then less than or equal to b. And this is the exact same thing as that. Let me just redraw it. So, let me draw our coordinates. Our coordinates right here. Our axes. So, that's the y axis and this is the x axis. So, when you evaluate r of a, right? That's our starting point, so let me do that. So, r of a, maybe I'll do it right over here. Our position vector valued function evaluated at t is equal to a, is going to be equal to x of a times our unit vector in the x direction, plus y of a times our unit vector in the vertical direction, or in the y direction. And what's that going to look like? Well, x of a is this thing right here. So, it's x of a times the unit vector. So, it's really, maybe the unit vector is this long, it has length one, so now we're just gonna have a length of x of a in that direction, and then same thing in y of a. It's going to be y of a length in that direction. But the bottom line, this vector right here, if you add these scaled values of these two unit vectors, you're going to get r of a looking something like this. It's going to be a vector that looks something like that. Just like that. It's a vector, it's a position vector, that's why we're nailing it at the origin but drawing it in standard position. And that right there, is r of a. Now, what happens if a increases a little bit? What is r of a plus a little bit? And, I don't know, we could call that r of a plus delta or r of a plus h. We'll do a different color. So, let's say r, let's say we increase a a little bit, r of a plus some small h, well that's just going to be x of a plus h, times the unit vector i, plus y times a plus h, times the unit vector j. And, what's that going to look like? Well, we're gonna go a little bit further down the curve. That's like saying the coordinate x of a plus h and y of a plus h. It might be that point right there, so it'll be a new unit vector. It'll be a new unit vector. Sorry, it'll be a new vector, position vector, not a unit vector, these won't necessarily have length one, that might be right here. Let me do that same color as this. So, it might be just like, just like that. So, that right here, is r of a plus h. So, you see as you keep increasing your value of t until you get to b, these position vectors are going to keep specifying, we're gonna keep specifying points along this curve. So, the curve, let me draw the curve in a different color. The curve looks something like this. It's meant to look exactly like the curve that I have up here. And, for example, r of b is going to be a vector that looks like this. It's going to be a vector that looks like that. I want to draw it relatively straight. That vector, right there, is r of b. So, hopefully you realize that, look, these position vectors really are specifying the same points on this curve as this original, I guess straight up parameterization that we did for this curve. And I just wanted to do that as a little bit of review cause we're now gonna break in into the idea of actually taking a derivative of this vector valued function. And I'll do that in the next video.