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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? •  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.
• This might be the silliest question. But won't the function fail vertical line test? • 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.
• At , Sal says.."The actual coordinate in R2 on the Cartesian coordinate.." what does R2 mean? • At he 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? • Why do we multiply with the unit vector? • Do the vector fields and vector functions mean the same? • 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.
• the graph drawn is not a function its a relationship...it gives 2 values of y for given x.
(1 vote) • 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? • 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.  