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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 9

Lesson 4: Defining and differentiating vector-valued functions

# 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.
• 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.
• At , Sal says.."The actual coordinate in R2 on the Cartesian coordinate.." what does R2 mean?
• It means the x-y coordinate plane. R for the real numbers, and 2 for the number of dimensions (loosely speaking)
• Up to ~ , 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.
• Why do we multiply with the unit vector?
• 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).
• 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?
• 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.
• I see the benefits of a parametric function instead of a "normal function". It gives you not just position but direction, it can limit the space you run through that function and it can give you time if you interpret "t" like that. However I dont see the difference between a parametric function and a vector valued function. In the parametric function points "appear" one after the other and in the vector valued function the points are signaled by vectors from the origin...so what? What is difference in the information that you get appart from a more comfortable notation? I see none...thanks for the help!
• Aside from semantics, parametric functions are the same as vector valued functions. For a normal parametric function (in 2d), you have 2 separate equations for x and y, x(t) and y(t). For a vector valued function, you have 2 separate equations making up your vectors, x(t) and y(t). The only real difference between the two is whether you have points on the function or vectors pointing to points on the function. Hope that helped!