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## Multivariable calculus

### Course: Multivariable calculus > Unit 2

Lesson 4: Differentiating parametric curves# Vector-valued functions intro

Using a position vector valued function to describe a curve or path. Created by Sal Khan.

## Want to join the conversation?

- I don't get what do the i and j unit vectors represent. I understand that if you don't multiply x(t) and y(t) by them, then r(t) won't be a vector. Do i and j represent in what direction function moves as you move on x and y axis?(7 votes)
- I and J are unit vectors. As an example in the 2D plane Sal used the unit vectors would likely be [1 0] and [0 1] normally as vectors they'd be written vertically, but bear with me. So, this basically maps to a coordinate pair (1, 0) (0, 1). Well, in vector math we can multiply these by a scalar and get new vectors. So... if we multiply a unit vector by a scalar we only change the base unit. In this case our vector [1 0] can represent the x-axis, the x-coordinate. If we multiply this unit vector by 3, for example, we get 3 * [1 0] = [3 0], or the point (3, 0). We can do the same for y. So, if we want the point (4, 5), represented as an equation for a unit vector, we get:

4 i + 5j, where i and j are our unit vectors for x and y, giving us:

4 [1 0] + 5 [0 1] = [4 0] + [0 5]

Now, when we add vectors we just add the terms of each corresponding part (so add the left part to the corresponding left part, i.e. 4 + 0 and 0 + 5

The new vector we get is [4 + 0 and 0 + 5] = [4 5], which represent our ordered pair and coordinate (4,5), Ta Daa!

Basically a unit vector is a simply way or generalizing a single direction in N-dimensional space. So, given an N-dimensional Vector we should have N separate unit vectors. Each unit vector is simply a separate "direction" in the N dimensional space, when added together, you get the proper direction/vector. I took some liberty here with my nomenclature, in proper Linear Algebra the notions of direction, vectors, etc are more well-defined. My intent here was simply to make it a bit easier to understand, not to make it more formal/rigorous.(33 votes)

- This might be the silliest question. But won't the function fail vertical line test?(5 votes)
- That's a good question, actually. Thing is, the vertical line test is useful when you need to see if the input (x value) is giving a single output (y value) or not. Here, x is the independent variable while y is the dependent one.

However, in vector functions, x and y are both dependent variables. t is the independent variable. So, if you had to do a vertical line test-like thing here, you'd need to test if one value of t corresponds to a single pair of (x,y) coordinates (you can't conduct the test like how you do it for scalar functions), which it does because if t is taken to be time, there's no way the curve is at two different points at a given time (It can return to the same point at a different time though, but I doubt you'll see such cases, as the curve would then cross itself over and that's not something we want it to do most of the times, especially when you eventually learn about Green's Theorem). So, this is the vertical line test equivalent for vector functions.(9 votes)

- At1:08, Sal says.."The actual coordinate in R2 on the Cartesian coordinate.." what does R2 mean?(4 votes)
- It means the x-y coordinate plane. R for the real numbers, and 2 for the number of dimensions (loosely speaking)(6 votes)

- At3:08he uses "i-hat" to signify the horizontal vector. In Algebra, when plotting imaginary numbers, "i" was the vertical component. I understand we are talking about 2 different things but why would "i" be used horizontally for one thing and vertically for another? Is this just an instance of "math vs engineering" notation?(4 votes)
- There's no particular reason. It's an accident of history.

If it helps, engineers will sometimes use j to denote the imaginary unit, so the notation aligns in that case.(5 votes)

- Why do we multiply with the unit vector?(3 votes)
- x(t) is technically a scalar, so we multiply by the unit vector (the vector equivalent of 1) to make it a vector component of r(t).(6 votes)

- Do the vector fields and vector functions mean the same?(3 votes)
- No, they don't. In this case you have a function whose domain is a subset of the real numbers. In the video, Sal considers the interval [a,b] as the domain of the function.

For vector fields, the domain is a subset of R^n and the image is also a subset of R^n. So vector fields take n-dimensional vectors as inputs and have n-dimensional vectors as outputs.(5 votes)

- the graph drawn is not a function its a relationship...it gives 2 values of y for given x.(1 vote)
- In this case we are not talking about y being a function of x.

If we were, you would be correct, but note! if we had x=f(y) that to could be the relationship here.

In this case the graph is a function of t.

Given a value of t, r(t) = x(t)î+ y(x)ĵ, a value for t gives just one value for r, so r(t) is a function.

You may wan to review:

https://www.khanacademy.org/math/algebra-home/alg-trig-functions/alg-parametric/v/parametric-equations-1(8 votes)

- Can these functions also be called vector-valued functions? That is the wording used in my textbook. If they are not the same, what is the difference?(2 votes)
- This is a very good question! A position vector (as opposed to a vector) starts at the origin and therefore determines a specific position in the region – i.e. a particular place represented by an (x,y) coordinate where that vector ends. A vector (non-position vector) does not. For example, the vector from P(0,0) to Q(1,1) is the same as the vector from R(2,1) to S(3,2) – both have the same magnitude and direction, but are in different places in the region.

A vector valued function (also called a vector function) is a function (not a vector) that outputs a vector, as opposed to a scalar or real value.(5 votes)

- Up to ~4:26, I think it's going to be easier to understand if you include the t-axis, even if it's just a quick sketch off to the side. As it stands, your original diagram for (x(t), y(t)) fails the VLT , and is going to confuse the viewer. In R3, of course, (x(t), y(t)) is a perfectly valid function -- which is the point. Parameterizing takes you from R2 to R3.(1 vote)
- We choose not to, because parameterization will have been well-understood by anyone at this point. Additionally, convention dictates that we don't show an axis for the parameter.(6 votes)

- Can anybody recommend a good book for multivariable calculus(for the purpose of question practice) as I do not find Khan Academy's assignments sufficient ?(3 votes)

## Video transcript

- [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.