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# Vector-valued functions intro

AP.CALC:
CHA‑3 (EU)
,
CHA‑3.H (LO)
,
CHA‑3.H.1 (EK)

## Video transcript

let's say I have some curve see and it's described it can be parametric parametric eyes 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 hey 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 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 this and 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 to 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 a 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 unfolded but it's hard to draw a bolt 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 position vectors 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 directions 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 say it's a position vector you're implicitly saying this is specifying a unique position in this case is going to be in two dimensional space but it could be in three-dimensional space or really even four or five whatever n-dimensional space so when you say it's a position vector you're literally saying okay this vector literally specifies that point in space so let's let's see if we can describe this curve as a vector a position vector valued function so we could say R of T let me switch back to that pink color well this K in green is equal to X of T times the unit vector in the X Direction the unit vector gets a little carrot on top a little hat that's like the arrow for it that just says it's a unit vector plus y of T y of T times J if I had if I was dealing with a curve in three dimensions that 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 well for any T and you know and still we're going to have T is 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 let's go X of a is this thing right here so it's X of a times the unit vector so it's really you know maybe the unit vector is this long it has length one so now we're just going to 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 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 kneeling it at the origin but drawing it in standard position and that right there is our 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 R of a plus h we do it in different colors 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 going to go a little bit further down the curve that's like saying the coordinate X of a plus h and y plus a plus h éeer it might be that point right there so we will go to it'll be a new unit vector it'll be a u a new unit vector sorry it'll be a new vector position vector not a unit vector these are don't necessarily have length one that might be right here let me do that same color as this so there 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 be these these position vectors are going to keep specifying we're going to 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 is going to be a vector that looks like this it's going to be a vector that looks like that let me I want to draw it relatively straight that vector right there is R our of be so hopeful you realize that look these position vectors really are specifying the same points on this curve as this original I guess straight up parametrizations that we did for this curve and I just want to do that as a little bit of review because we're not going to break in into the idea of actually taking a derivative of this vector valued function and I'll do that in the next video
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