Review of Vector Functions – Concept Check by @mes on 3Speak
In this video, I go over the concept check review from the end of the vector functions chapter in my calculus book. I cover vector functions, space curves, their derivatives & integrals, curvature,...
Tags: #math #calculus #science #vectors #chessbrothers #threespeak #scifimultiverse #vector-functions #education
Now let's dive into the review, so concept check. And then, yeah, so these are a bunch of these questions here. And again, actually, for completeness, I'm going to read all these questions out, and just in case you want to solve them yourself, and I put them all in order, and then I'll go over one by one, just in case you want it to actually try to solve this yourself. So question one, vector function calculus, what is a vector function? How do you find its derivative and its integral? Then question two, space curves. What is the connection between vector functions and space curves? Question 3, tangent vector. How do you find the tangent vector to a smooth curve at a point? Then, how do you find the tangent line? The unit tangent vector, question four, differentiation rules. So if you and V are differentiable vector functions, C is a scalar, and F is a real valued function. write the rules for differentiating the following vector functions, and we have these right here, and this is the addition of these vectors, and basically take dirt of that, and this is constant times a vector, and then this is a function times a vector, and then dot product of two vectors, and there's a cross product, and there is a vector. a function within a vector, and then how do you take the derivative of that? And then question five, length of a space curve. How do you find the length of a space curve given by a vector function R of T? And then question six, curvature, what is the definition of curvature? And then we're asked, write a formula for curvature in terms of R prime and T prime, like that. and this is the derivative of the unit tangent vector, and then we're asked write a formula for curvature in terms of R, the derivative R, of T there, and then D is write a formula for the curvature of a plane curve with equation Y equals FX, and then question 7, unit normal, binormal planes, and oscillating circle, so we're asked right formulas for the unit normal and binormal vectors of a space curve, R of T, then be what is a normal plane of a curve at a point? What is the oscillating plane? What is the oscillating circle? Then, question 8. Velocity, speed, and acceleration. How do you find the speed, the velocity, speed, and acceleration of a particle that moves along a space curve? Then, rest, write the acceleration in terms of its tangential and normal components. In question nine, Kepler's Law, state Kepler's Laws. All right, so I read all those, and then, yeah, if you want to solve those. on your own, feel free to do that, and then compare it with the answers. So, anyways, a solution to question one, so vector function calculus. So what is a vector function? And then we're asked, how do you find its derivative and its integral? All right, so let's take a look at the solution. So a vector valued function, or a vector function, is a function whose domain is a set of real numbers and whose range is a set of vectors. So essentially, you put in a, yeah, put in a real number in the input, and then what you get out of it is a set of vectors. All right, or you basically get a vector. So you put a real number in, you get a vector. So in general, a vector function in 3D space can be written in component form, just like equations for lines. That is, any vector function is of the form, where the parameter T typically denotes time. And yeah, that's why typically, just pick it. Basically, would have a vector function in 3D, and again, it's most common, just 3D is how you can have direction and magnitude, and so on. So we're going to get R of T is equal to, so the function F of T, and then this is the times by the i standard basis vector or the x component and then we're going to get a function for on the component function for the is the j term this the j or the y axis or the y the y standard basis vector and now we have the vertical component this is going to be h of t and this is going to be k standard basis vector and that's for z-axis and so on And here quickly, recall from our earlier video on vector functions. If you click on the, this is the video notes, and you scroll to the bottom, you get the thumbnails for each of the individual sections of that giant video and the playlist and so on, but basically I put the thumbnails all at the bottom there. So this thumbnail I made, so RFT equals to basically have the F of T, GFT, HFT in this triangle format. this triangle or bracket set up there, and it's the same thing as writing this with the center basis vectors. And now I illustrated the real valued functions and then vector valued functions. Just illustrate that you have a domain, you put individual input, and you get an individual output in the range. And then in vector value functions, you put individual input, and then you get vector value set up there. So you get vectors, for example, you put an apple, you get these three things. you get apple, strawberry, banana. This one, he's put apple, you get a strawberry. I just wanted to illustrate that. But in general, yeah, for vectors, we're dealing with a real number, so not imaginary or an apple. Just a real number, typically time for T, and then you get just a vector, typically two or three inputs like that. All right, let's go further. So to differentiate the vector function, we differentiate each component separately to get, and we just take, again, take derivative of this, each component. So the derivative would be R of T, R prime of T is equal to F prime of T, and then this is the I vector, plus G prime of T, and now this is going to be the J vector, plus, and then this is going to be H prime of T. This is going to be the K vector. And now to integrate the vector function, we integrate each component separately to get, and then basically do the same thing, but in reverse. So we do take the integral of, so the integral of this R vector, R of T, DT, now this is going to be the same thing. You're going to take now, I'll put this a bracket there, integral of F of T, DT, and now put this all in this I component, and then plus or do the same thing for the J component. We're going to do it like this. Integral of G of T, DT, and then this is going to be, this is the J component. Likewise, we're going to have a, right here, integral of H of T, DT, and now this is going to be K component. And yeah, likewise, recall my earlier video on derivatives and integrals of vector functions, there's a link to the video notes, and then scroll to the bottom, you'll see this thumbnail I made for that individual. I think this is for the, it might be for the full video or the one of the sections of video. But anyways, so recall, here's a thumbnail, so if you have, this is the space curve there, is a vector function like that, and then basically when you take the... Yeah, take the derivative r prime of t like that, it's going to be the tangent vector, and then this is going to be the, yeah, this can be the secant vector, basically this to a point like that, and then if you use the definition of derivative, and then as the limit, this goes to the difference, this H goes zero, you become this derivative set up there. So that's the derivative there, and I also showed that you can take derivative by components, so we get like that. and then the integral is going to be, well, just integral body components like that, so integrate. So again, make sure to watch that video. Again, this video is going to be review and so on. So a lot of this is I'll be just reviewing and also linking back to the early videos, which I cover in depth. All right, so now let's take a look at solution to question two, space curves. So question two was asked, what is the connection between vector functions and space curves? And let's look at the solution. So a space curve is a curve in space. There's a close connection between space curves and vector functions. Specifically, we can determine a vector function, which traces along a space curve C, provided we put the tail of the vector at the origin, at the origin, so they are, or vectors of the order, if we're doing many vectors, so they are positioned vectors. Likewise, any vector function defines a space curve. So basically, if you have the origin, you put tail out there, then you put this, and this is right at the... the space curve, and as you go around from it, and as you vary this with time, you can again get a curve in space. So, yeah, so for example, if we have a vector function, R of T equals F of T, and then I plus G of T, I, J, plus H of T, and then K, the center-based vectors, and we vary T, The position vector RFT traces out the curve C. So see my earlier video on vector functions and space curves. There's the thumbnail. There's a link to the video notes. And there's a thumbnail. And what you have is, again, there's the curve, space curve. And basically, every single vector function defines one, or it traces out one. So any vector function defines a space curve as you vary T. As you vary this, this is going to go all the way around it. So initially, it's going to be here at different... T value, this is T, a different value, and it's going to go and spiral around, depending on what happens to the function. Yeah, and yeah, that's how does vector functions like that. And remember the X component, this is just X component, X equals F of T, Y equals to GFT, and so on. So it basically plots out. the distance up to the x component, and then distance of the y, and then you have the distance of the z, and put them all together, you're going to get a 3d space like that, a 3d cube, and the only point that fits all of them is going to be a point like in space somewhere here. It's fascinating, fascinating stuff there. Alright, and now let's take a look at the solution to question 3, and that's the tangent vector. So how do you find the tangent vector to a smooth curve at a point? And then we're asked, how do you find the tangent line, and then the unit tangent vector? Alright, let's take a look at the solution. So let's see be a smooth curve with position vector R of T. Then basically, the tangent vector is going to be R prime of T. It's a basic derivative of this one. And then the unit tangent vector is basically just a tangent vector, but it has to be magnitude 1 or length 1. So the unit tangent vector, this capital T of T, a small case T, equals to just a derivative. So the unit tangent vector, divided by the length of the unit tangent vector. So that, just so that this entire vector, we normalize it so that the length is going to be one. So then the magnitude of this. That's all it is. Alright, so now going further, so I just quickly recall from my earlier video, again, there's link to the video notes, and then you'll see all these thumbnails, scroll down to the bottom, and you'll see that, and then, you know, recall my earlier video on the definition of derivative of a vector function, and basically you take the, you take the secan vector, divided by H and then take the limit as H apart 0, and basically, yeah, we just get our derivative there, that's a tangent vector, and then the unit tangent vector is this right here, so R prime divided by, yeah, R prime of T, divided by the magnitude of R prime of T. All right, so now recall also, my earlier video for the vector equation of a line, and that's because we're going to go look at how to find the tangent line. All right, so recall the vector equation of a line, so we have a line like this, L, and then this is going to be, this R vector, it's going to be denoting the line, so at all points of it, and then it equals to basically just the initial line, initial vector, R0, so at a point A there, or actually at a point, just at a point not there, or whatever, X not, or T, yeah, so then you would plus T times by a parallel vector, so this is the parallel vector to this one here. All right, and then you add those up, and then basically you're going to have this, basically, this plus A, where A is equal to TV. So that's the equation of a line, so basically just using vector addition, so add them up and equals to this vector. So this plus this equals to this one here. Alright, so now that brings us to how to solve the vector equation of a tangent line at t equals a. So let's just draw this out to make it a bit nicer. So I'll just grasp it a bit. Yeah, just because I actually haven't, I didn't do this in the earlier videos. So if we have this, like, there's the origin, this is the Z, this is the Y, this is the X. Or let's say we had a curve like this. Here, I just fix that up. So let's say this is C, that's our curve. And now let's say we have our R of T. Yeah, let's say this is our R function, but this is at the value A. Yeah, I'm just going to put this as A here. That's just the one we want. And let's say the tangent vector is going to be parallel to this, I mean, it's going to be tangent to this curve. So that's going to be R prime of A like that. Alright, and then the line, yeah, I'm just going to draw this in dash lines, actually, dot, dot, dot, dot. This is going to be our line all the way across, and that's going to be our line L. All right, here, I just fix that, I've moved the L just over there. So that's what we have there. Now, if we look at our vector equation of a line, basically, we have the vectors equal to the initial value, plus, and then a parameter T times it by... It's just a parameter, because we already have T for R, I'm going to put it as a different one, and then times it by a parallel vector. This is parallel, I should put a parallel lines there, maybe, maybe later. Anyways, that's parallel vector there. So what this means as well, in our case, this is a parallel vector to the line. This is parallel to it already. So then we're going to call the line L, so this is their line. I'll call this the Solution manual or the solutions on the Quizlet website. puts it as s vector, and then the parameter is going to be u instead of t because t is already taken for the r so this equals to the initial value so the initial vector r of a plus and then plus well u and then the tangent vector this is just the u parameter times it by r of prime or prime of a like that. And here, just move this around there, just put an arrow there. So there is our vector equation of a tangent line at the point T equals A there, yeah, so that's fascinating stuff there. All right, and now let's take a look at the solution to question four, and this was going over the differentiation rules for vector functions. So this one states, if U and V are differentiable vector functions, C is a scalar, and F is a real valued function, write the rules for differentiating the following vector functions. So we have A, this is a dot, I mean, this is a vector addition, and then B is the scalar times it by a vector, and then C is the, just a function times it by... a vector, and then we have dot product and cross product, and then when we have a function within a vector function like that. So let's take a look to solution. So these are the differentiation rules for vector functions that I went over in my earlier video. There's a link to the video notes, and is the screenshot. Again, scroll down the bottom, you'll see all these. There, so we already have this. This is the first one is the addition one. So basically, when you take the derivative of the two vectors that you're adding, the answer becomes basically just the, yeah, it's the same as taking the derivative of each separately. And then likewise, you know, likewise, if you have a scalar, you can just take that out of there. So, or just basically take the derivative of the function, and the scalar doesn't do anything. Then if you have, then you have a function like that, what you do is basically use the chain rule. So you'll first take the derivative of the function times by the vector function. and then the function times it by the derivative of the vector function. So again, it's like the chain rule there. And then the number 4-1, this is the dot product, and it's very similar to this setup here, using the chain rule setup. So derivative of the first function, dot product with the second function, second vector function, and then you'll have plus u of t, and then dot v prime of t. And then for the cross product, it's the same thing. It's the same exact thing. It's the derivative, cross with the other vector function. and then we'll have the vector function cross with the derivative of the other. So it's very fascinating. So then we have the function one. We just use the, yeah, this is the chain rule, actually, this is the chain rule. The other one was basically product rule and so on. Yeah, these are product rules. These are the chain rules. So the chain rule, basically, it's the same as taking, you take derivative of the vector function, and then you multiply by the derivative. of the real valued function inside. And also, for completeness, the derivation for each rules illustrated in the sections, video, thumbnails below, and again, all these thumbnails are in this video link, or the notes link, and scroll to the bottom. So here is the solution to A, this is the vector addition one, and again, using just the definition of, or, not definition, but basically the component, the derivative of vector function to buy components. So you put this inside, then you're just using, just, then you just use basic real valued functions, properties, and it becomes the exact same thing. Just take derivative inside like that, and you add them up. And so there's the rule there, and then for part B and C, this was, if you have a real value function here, same thing, same thing, would you end up, just do that, do it by components, take derivative by components. and then you're going to get this set up here, and then the special case where the real value function is a constant, then what that means is if it's a constant, the derivative of a constant is zero, so then this vanishes, yeah. So you have this, this part vanishes, you just get C, uprime. All right, and then the dot product, same thing here, you just take dot product like this, do by components, and so on. Yeah, but in this case, you actually multiply this dot product out, you get this set up and you can turn it into summation, and yeah, it ends up becoming just this. So again, go and make sure to watch that video. But again, the quiz wasn't asking for derivation, it just says, what's the answer? So, and then there's derivation for this one. It's a cross-product, again, the same thing, product rule set up. there, and this one, if you just do it by brute force, and then you're going to get, you're going to get some cancellations, and then put this all in together, you get the exact same thing. And then, yeah, what ends up happening is you basically have the definition within this expansion when you expand it out, and do the dot part, I mean the cross part. Anyways, and then for part F, you could take a derivative, and then you get, this is the chain rule there. Again, do it by components. and you end up getting the same thing. So the idea of doing it by components basically means you could now use the properties of real valued functions, and then what you end up having is exactly the same analogous thing for the vector function. So basically derivative of the vector function, and then multiply by the derivative of the real valued f function there. Alright, and now let's take a look at the solution to question 5, and this looks at the length of a space curve. So, Russ, how do you find the length of a space curve given by vector function R of T? And the solution here states, well, the rate of change of the distance with respect to T is. Let's do this first. It's a quick derivation from that Quizlet website. So DS over DT. So this is the rate change of that tangent vector. and then, but you take the magnitude of it. So you take the magnitude of the tangent vector, because again, if you just…