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Review of Vector Functions Chapter – Concept Check & True-False Quiz by @mes on 3Speak
In this video, I go over the concept check questions and true-false quiz as part of the review on the vector functions chapter from my calculus book. There are 9 concept check questions and 14 quiz...
Tags: #math #vectors #calculus #education #quiz #chessbrothers #threespeak #scifimultiverse #science
Hi, it's time for another math. Easy solution. Today I'm going to go over further into vector functions and now look at the review and true false quiz at the end of the chapter. Yeah, at the chapter that basically reviews everything there. Before we get further into it, note as always, if you don't have time to watch this whole video, you could play this video at a faster speed, top secret left hack, your brain gets used to faster speed. There's a link to my tutorial there, and you could also download and read the notes. It'll be there in PDF format, as well as an article format on the Hive blockchain, and the link to all this will be in the description below, and also you could watch a video in parts. All the time stamps of all the parts are in the description. You could click there and even skip this intro if you want and go to any section you want. And also, I'll be uploading a video section's playlist in time and basically clips of this long video and then put it in a playlist so you can go specifically to the topic and so on. Except this video is going to be true falls and review and quiz and so on. I might not do it for every question, so I might just do it, just a bulk one. just because it is a review. Anyways, let's go further, so calculus book reference, a note that I mainly follow along the book, Calculus, early transcendent, 7th edition by James Stewart, there's a link to that, and I use the following solution manual for this chapter, and also note that I use the Quizlet .com 30-day free trial for the solutions to the concept check. There's a link to that, and also note in some earlier videos I use the 6th edition, is the 7th edition. and that's the six I used earlier on anyways let's continue further so let's look at the capitalist book chapters of the hive notes and sections playlist for each video of this chapter are listed below there's a link there's a playlist the YouTube playlist to all the all the sections of the of the chapter, and there's the hive notes, and there's the sections playlist of each subchapter there. And the current one we're on right now is this review right here, and there's the playlist link, and then we're going to do concept check and true false quiz in this video, and then Problems Plus, and in a future one. So stay tuned for that. Also, these links are also on MES links, and you can go there. So you can go here, and all the links will be in here. If you go to the, you can do this droPeakDown setup there, you can go right here, this opens up everything. That's interesting, I've got to fix that up, but anyways, so if you click on tutorials right here, and then there's the infinite sequences in series, and basically if you drop down, there's vectors in geometry of space, click down, there's the vector functions. and so on, and then I need to add the review, and so when I'll be adding that once I'm done it. All right, so epic stuff. All right, so let's go further, so topics to cover, so the concept check, there's going to be a bunch of these questions. I'll read these out, and then again, you can just skip this in the times times below, but I'll be read these out. and so on so concept check question one will go over vector functions uh yeah vector function calculus and then look at question two space curves question three tangent vector question four is going to be on differentiation rules of vector functions Question 5, length of a space curve, question 6, curvature, then we'll look at unit normal, binormal planes, and oscillating circle, and then velocity, speed, and acceleration, and look at Kepler's laws, again, this is, just these topics to cover the titles, I put them in there just because it's a good review in the end of itself, and then the true-faults quiz, we'll look at the vector equation of a line, parabola, line through the origin, differentiation by components, derivative of cross-product, derivative of cross -product, derivative of a vector, curvature, binormal vector, and inflection, points and then curvature of a straight line and then constant magnitude of a vector and then look at space curve on a sphere and then the osculating circle, and then parameterizations of a curb. All right, so it's enough of that, so it's about three minutes in or four minutes in. So 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, 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 three, 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, and then the second 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 seven, 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 B, what is the 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? And 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, a solution to question one, so a 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 real number in the input, and then what you get out of it is a set of vectors. Basically, you 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 it typically is 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 is the 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 want 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 a 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 a 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. Alright, 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 detail of the vector at the origin. at the origin, so they are, or vectors, or vectors, 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 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, J, plus H of T, and then K, the center-based vectors, and we vary T, the position vector R-T 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, 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, then, this is just the X component. X equals F of T, Y goes to G of T, 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 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 then 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 as 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 if 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, these are 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 fixed 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 R of, this is our R function, but this is at the value A. Yeah, this is 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. All right. 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, 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 up parallel lines there, maybe, maybe later. Anyways, that's parallel vector there. So what this means is, 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 solutions 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's 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, R 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 again fascinating stuff there. Alright, and now let's take a look at the solution to question 4, 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 then when we have a function within a vector function like that. So let's take a look to the 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 to 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 uh just the uh yeah it's the same as taking the derivative of each separately and then uh likewise you know likewise if you have a scalar you can just take that out of there so or just take just basically take the uh derivative of the function and the scalar…
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