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Find equations of the trajectories of the following vector fields: (2y, (39, (y,2) (21,y) 1,TV) (cosh = cos y, sinh = siny) U - (T,yln |yl) (v,2,-V) V = V, I+V, 1)....

Question

Find equations of the trajectories of the following vector fields: (2y, (39, (y,2) (21,y) 1,TV) (cosh = cos y, sinh = siny) U - (T,yln |yl) (v,2,-V) V = V, I+V, 1). 17_ Sketch the trajectories ofthe vector fields in Question 16(a) to (e) roughly; indicating their directions.

Find equations of the trajectories of the following vector fields: (2y, (39, (y,2) (21,y) 1,TV) (cosh = cos y, sinh = siny) U - (T,yln |yl) (v,2,-V) V = V, I+V, 1). 17_ Sketch the trajectories ofthe vector fields in Question 16(a) to (e) roughly; indicating their directions.



Answers

Trajectories on circles and spheres Determine whether the following trajectories lie on either a circle in $\mathbb{R}^{2}$ or a sphere in $\mathbb{R}^{3}$ centered at the origin. If so, find the radius of the circle or sphere, and show that the position vector and the velocity vector are everywhere orthogonal. $$\mathbf{r}(t)=\langle\sin t+\sqrt{3} \cos t, \sqrt{3} \sin t-\cos t\rangle, \text { for } 0 \leq t \leq 2 \pi$$

How this problem gives us a position. Flexion was this reflection Are Electrabel are off t use three components. So we are in the three dimensional space. First Components three, sign the, uh, second components. Go scientist E on the third is for for see for the between zero. Well, we need to find Does the trajectory lie on a circle? Ah, circle centre at the origin has an equation like this. So we will be squaring anyway. Second If so, we need to find the radius on a proof or test that the velocity victor off t you or struggle to the position. Victor. So in three dimensional space would say this is one of the victory are the are the the beginning point is that the origin on the end point of each of these is in a point that has scored six. Once the eggs war with the to check whether these points lie on a circle off a circle of radius, this would have to be constant, right? So with square X squared plus y squared, squared equals the square of this is foreign Group C. What's the square of this? Is the 25 close sine squared t on the screw. This is six green, mean great t like terms to open here, So we have to 25 sign screw t plus these. Thank you. Five through the throughout. All right. And lift with fundamental Pythagorean identity off trigonometry. This is one. So this is 25. This is five squared. So the answer to the first question is yes. Uh, circle you've for the two dimensional or for the three dimensional. I'm sorry. I should have said great sphere in three dimensions. A circle of the secretion. That is it. So you the victory live trajectory live. Oh, a fear. Thank you are a, um here. We're radio. Uh, okay. All words to find the trick. With the victors off, velocity and position are orthogonal. First to find the victor of velocity off T is the derivative off the position off the which you wait. Go back to each of these. Compose on, uh, find the derivatives of each of them. So good. Route off. Sinus ghost science of the route. This would be three. Go 70. This would be the root of coastline is minus. Scientist B minus five. Sign on this before go. Sci fi The velocity Victor is three. Go sign T Uh, months. Sorry on for co sign the times. Uh, looks perfect. Are is a plane by multiplying the first components off the music. Sure. On the both fractions. Roads three times three is nine signs. 70 mine. Fine. Okay. Murder 4001 2 25 Fine. He goes and four times for 16. Sign all together. Like terms, three life terms. So we have minus 25 we have plus line. Plus it. Plus 25 find the coastline, which is zero. So this is that the victors off the end, uh, position are or funnel, huh? But if it I hope it helps.

Hello. We are given a position function, Uh, in the three dimensional case far too. Uh, position function is a victim functional. The in component form for son. The I am component to profile. Fine on in through all for the zero to part two questions. When is it was a lie on a circle? If it does, What is the radius on? Secondly, Ah, we have to show that the the lost Victor is or for Google to of the position Victor, which is find with perfect being zero. Okay, uh, in the next four playing, if we have Victor r t the terminal point of each Vitter's some point x y So we take pick some or to be these two And the trajectories think to be wherever these victors are pointing for values of teak from your part. Okay. Circle scented at the radius have, uh, an equation experience. Plus wife's critical R squared. And after his job, used to find out with a, uh, the 0.64 satisfy this thing here for squaring ex clearing. Why? We have things. 16 Fine square, t. But four growth hormone square. Um, we can run the 16 sunscreen is 12. Find where the four run, please. And we had before Coastline Square here in order to extract for a factor of four here, this will be sine squared, plus four squared. See, uh, through have fun. Uh, Pythagorean, actually, this was a plus in this sine squared C plus co. Sanskrit sees the fundamental Pythagorean equation. I don't see Sequels. Well, so ex script before square? Yeah, not confident changes depending on the so, Well, fun screwed the bluff four, which is not equals some radio for this here. You what is probably a little but it's not a circle. So the answer. Food question. Well, you know, the trajectory is not on a circle sense that the origin the second part used to find First to find the velocity victor, which is defined as the the relative, all the position. Victor, uh, we find the derivative by finding the riveters off component functions. So the fourth sign T is equal to four. Go society. Uh, the other go sign thing is two times minus five, which is Marlon too far. I think, Uh, actually, the second part isn't the compulsory, because the second part is if the trajectory allies in a circle, then find the radius. And if so, check that the This is unnecessary. The answer is, uh, the trajectory. It does not lie. I never have it. Hope it helps, but

Oh, in this problem were given a position function are tee is Victor function given in component form eight. Go sign a to three on the single Guarantees eight. Sign Oh, to t already for the between zero and pie between zero and by all right. The first thing to notice is that Ah, uh huh. This is in two dimensions. When we have a victor off position, then he starts of the Ergin on terminates at some point. Uh, exploit. Right. So I think this is six. This is why on DWI Suspect, actually, we have placed youth. Does this point through all these points where these vectors are 3.2? Do they belong to a circle of some? Well, some radius Felix Square eight growth sign T on a fine to see a fever, we get, uh, 64 go, son squared off to T plus 64 off course. Sign square. Uh, apartment find spread the effect throughout. 64 left with ghost sine squared to see but sign script seen Addison parentheses. This is the fundamental By figuring identity, this is equal one. So this expression here furniture these evil 64 which is a square, therefore. Ah, the end points off the position. Victor, lie on a circle, but have, uh, the equation excreted widescreen equals a squared, which is a circle off radius four lying on the urge. Okay, for the first quarter question is to determine whether the following trajectories lie either in a circling. Uh, this is the case. A two dimensional r squared on. The answer to the first question is yes, The trajectory, Um, the position, like there is, in fact, on a circle that has radius of a in the plain X lightly. The second thing is, we need to show that the position vector on the velocity vector. And what is velocity velocity? If the the velocity is the derivative, All the position flexion on the off the position Faction is the the off comparing a little further side north here on find what is a derivative of eight co sign to see. All right, this here is a composition of factions. The route is going to be The sign is minus fine off to sea times of the route off to t, which is to well together. It's minus 16 fine off the team. All right, So that is the foot component off the velocity. Uh, on the second with derived eight. Sign off. They timed the game. It's a complex attraction for the root of scientists. Ghost time off to think times the root of all on this is 16. The co sign to think we need trick. You've are times the equals zero because this is the conditional or finality. Uh, just make some room here and will continue. Okay, the product off our and the is Oh, just start writing it out. Uh, are off t times the off t begat product is Did I Soon. Ah, 16. Go side to think. Okay, that is the victor off a little steam. And here is the for all, uh, position. The first component. Times first eight co sign tee times My 16 fun. Good thing way have second component. Plus second component of second components is a fine too burned 16. It I'm 16 is 144. Uh, scientific. Oh, scientist percentage. See spots. 144. Scientific fine to sign. Ah, like terms. 144 144 0 Therefore through things. Uh the trajectory is on a circle Sen through the yours. Radius eights on. Secondly, uh, the velocity victories or a signal to the, uh, position with and that completes exercise hope. Help the

So again we have to solve. We have to find if our any trajectory RFD, which is given is are after is given is sign off D cause sign off. T and co sign off the lie in a circle Our sphere So we can see this is a three dimensional object X y z Three components are given. Okay, Once we have the X Y c it definitely it will never lie on a circle because circle is a two dimensional Jer. It lies. It may or may not bite disappeared. Okay, for a sphere, what we have to do We need to try to write it in the standard farm. All we need to calculate the distance between this point and there a region. Okay, once we will calculate the distance, it will be signed. Scare off t plus cause I'm scared after bless cause I'm scared after is equal to sound scared. Plus cause I'm scared from treating a metric ardently. It's 11 plus goes and scared, which is not equal to our off scare or any. A constant, you know, it's not equal to a constant. It should be able to constant. Then we will say so. It Zeal. Ionosphere. This implies I are off key. Okay? Does not lie honest. Severe. Yeah. This is the first argument. Uh, the other argument will be white. It's not like in a circle, because it's a three dimensional object. It's a three d vectors, the sim ply. It will never lie on circle. What circle is a two d two dimensional object? Okay, it is not like then we can not guilty radius the second part. And we don't need to check the r technology, that's all.


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