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22 rt) = 42 and (24 sin [,24 cos (,10t) unit tangent vector 'N= and the principal unit normal vector for the following parameterized 3 Verify that...

Question

22 rt) = 42 and (24 sin [,24 cos (,10t) unit tangent vector 'N= and the principal unit normal vector for the following parameterized 3 Verify that

22 rt) = 42 and (24 sin [,24 cos (,10t) unit tangent vector 'N= and the principal unit normal vector for the following parameterized 3 Verify that



Answers

Find the unit tangent vector $\mathbf{T}$ and the principal unit normal vector $\mathbf{N}$ for the following parameterized curves. In each case, verify that
$|\mathbf{T}|=|\mathbf{N}|=1$ and $\mathbf{T} \cdot \mathbf{N}=0$. $$\mathbf{r}(t)=\langle 2 \sin t, 2 \cos t\rangle$$

I guess we are. G is equal to to your sanity, Grady. Inside of tea to 70. Okay, we need to find our unit 10 directory. So take note of of Artie, That gives me negatives to sign of tea. Three and two costs energy. Now, let's take the magnitude of our primacy that gives me describe it of 13. So our attendants Bechler into culture. They have to sign of cheap gray and to co sign up, see all over Mrs Corbett of 13. Okay, now we're asked to find that units normal vector. So we need Thio. Take derivative of T. That gives me need to over 13. Garrett of 13. Co sign of tea. Sterile and negative to overdo Gene spirit of 13. Sign it. E No, we need to take the magnitude of that. Okay. Square roots of this holding squared. Plus, that morning squared that simplifies to this to screw whatever Tune over the Tyne for a normal rector. Know what unit vector is equal to? Um, what do you say I gotta to over 13 square bit of 13. Cool sanity. Zero and two over 13. Screw it. 13 sanity all over to scare a little 13 over the city and that simplifies to anarchy is equal to negative coastline of tea zero and negative side of T. Okay, now we're asked to find our final affected, but I've I know by normal, by normal I know that there is B a t is the cross product that tea cross and so across them we get This is it's a negative to over 13 screw. It's 13 sci fi. They're all oh, you know, three square with the other team trumpeting and two over the team spirit of 13 cool side of t crossed with negative coastline of tea They're all in the society. You take the cross product of these two. You see that this is equal to I got a three over 13 square root of 13 sign that negative to overthrow its energy and over the Tyne and three over 13 square root of 13 cool society

All right. So here we have thes privatized crew on. We're trying to find its unit Tanya Vectored and unit Normal Vector. So the first step is going to find extra motive. That's when it did. You ever have divided by the magnitude of the distributor. So the drivable distance it's going to be our penalty is equal to the derivative of each team separately. So did your beautiful four seventies for coal. Sign off key. The durable foreclose NT is negative for sign. Oh, I see. On the derivative of 20 is just 10 now into finding its monkeys. You, the magnitude is going to be our penalty you see equal to that's Beirut off each of these things square. And then we're gonna combine them. You'll see what I mean. So this is going to be 16 Coastline squared, cause for concern. T since forecasts nt is assisting coastline sports 40 And then we're going to wired negative forces. Negative. 40 16 on sign Teton Santee, Sign square. See? Now we have these 10 so 10 times. Sandy's just 100 Now we're gonna factories, it seems from each other. So notice how we call us. It's team right here on each one of them will benefactor it out. We're gonna be left course nt causing square t a sign. It's where t Hi disease plus 100. So this thing right here, cause it's 40% authorities attribute entity, so it's going to simplify into one on 16 times one, It's just it's team. So we're gonna have 16 1st 100 and then this is going to be 160 on This is just four times 29 inside this Beirut so we can take out the four. Just be left with two screwed up stunning night. Now we're going to, uh, put in his nominator, put the substance human reader like this. So we have the tendon vector off. The is equal to four. Co sign home on negative four. Sign off D. Okay, now we're going to Dubai. This by the magnets, you futures to spare room 29. Okay, great. Now notice how all of these lumbers coefficients are even which means we can factor it out to. So if we do that, let me delete this. Yes, on this. So we're who factor out the to the tennis, just turning to five. The's far is gonna train to to something with these four. Now we have a two indifferent are now based. Do on these two, they're gonna cancel out there like that. We just find or Italian banker Not within the city. Sanya factor to find a normal vector unit. Normal vector. So we're gonna take a distributor which is still going to be so she promised e is equal to 1/29 because that's a costing, so it doesn't change times out of pieces. So did you reveal to cause NT is negative to sign t on the derivative of this is negative two co sign t on that real five issues zero because five is a constant. Now I want to find a magnitude of this So the magnitude is going to be so the penalty. We're finding its my mind too. So we still have while always any night on then times the square root negative to negative two since number two some 70 square used to us for sine squared. Uh, 17. With this one, it's it. Instead, it's causing because the native two sons aged to sentence to a positive for we'll have us for. Oh, sorry. It's where as your times he was still cereal. So we just want to leave it like that. Okay, cool. So this is the magnitude. Now we're going to divide this by this. Now, notice how this is a fraction, right? So we're dividing this too. These two fractions are going to cancel out, right? So, what do you want? Anything like this? Just forget about everything that's over here on this is going to be on a normal back here. So everybody is equal to the care off. Negative to society. Home a negative too. Hold 70 Coma zero over. Now Notice how? Because of the same thing with the last time we just talked about the four on the science for plus persons 40 transit to one. So basically, I was gonna be left with on the coefficient. Is this brutal? For it's as we found out before this aspect of t plus causes Scrotie chances to one. Now, of course, Scooter for is just to those it's too. Now we continue to fight this even more because notice how we have a two right here. That's right here. This is just here So we contact our this year from each. So they say. So this is gonna be negative. One negative one. Don't tell it to right here. And we got to write. You write. So these two are gonna cancel that? I'm just on a recent, so it always more space. All right, let me circle this. Oh, of course. We don't need this line anymore as we're not dividing by any. So this is sort unit normal vector. Ah, now we could Who? The disease. The right answer By taking the magnitude of each one. Uh, this is what I mean. So basically, when you cut them by into the unit on your vector, it should be equal to the magnitude of the normal vector. They should be the same. So take the magnitude of this on the magnitude of this and see if they're the same. So first of my film, this is going to be one over this period of time. Nine. That's fair. Moves to cause anti times to cause entities just for call sign. It's where on this one is probably negritude. They get up to these positive for on scientists. Um, 70 East sign square see from five times five piece. Just 25. Okay, great. Now we can see the fighters were gonna back up before causing square plus course and square. I mean, close in support of people. Science, cruelty. It's just one. All right. So we have the course asperity. Well, science broke t before. That's what we d I forgot the teeth on 25. So this is off course. While I'm over this period of 29 we have four times one class 25 four plus 25. He's just 20 night like that for me. Put it like this. Now we have 1/39 times. Whatever. This fruit of 29 times this proved of 29. So this is going to cancel another. Well, that foot is one. Now, we're gonna do the same thing with enough tea, so we're gonna take its magnitude, which is negative. One sign. Lt those negatives wants. Lt is just sigh square. See on something with the course that's going to turn to course. Science parity on your hands here is just to so the distress into the spirit of wine that is off course, just one. So notice how this one give us? Why this one careless one? Which means that this is true. The monitor of tease people to the mind and which it means that this is right.

All right. So here we have thes privatized crew on. We're trying to find its unit Tanya Vectored and unit Normal Vector. So the first step is going to find extra motive. That's when it did. You ever have divided by the magnitude of the distributor. So the drivable distance it's going to be our penalty is equal to the derivative of each team separately. So did your beautiful four seventies for coal. Sign off key. The durable foreclose NT is negative for sign. Oh, I see. On the derivative of 20 is just 10 now into finding its monkeys. You, the magnitude is going to be our penalty you see equal to that's Beirut off each of these things square. And then we're gonna combine them. You'll see what I mean. So this is going to be 16 Coastline squared, cause for concern. T since forecasts nt is assisting coastline sports 40 And then we're going to wired negative forces. Negative. 40 16 on sign Teton Santee, Sign square. See? Now we have these 10 so 10 times. Sandy's just 100 Now we're gonna factories, it seems from each other. So notice how we call us. It's team right here on each one of them will benefactor it out. We're gonna be left course nt causing square t a sign. It's where t Hi disease plus 100. So this thing right here, cause it's 40% authorities attribute entity, so it's going to simplify into one on 16 times one, It's just it's team. So we're gonna have 16 1st 100 and then this is going to be 160 on This is just four times 29 inside this Beirut so we can take out the four. Just be left with two screwed up stunning night. Now we're going to, uh, put in his nominator, put the substance human reader like this. So we have the tendon vector off. The is equal to four. Co sign home on negative four. Sign off D. Okay, now we're going to Dubai. This by the magnets, you futures to spare room 29. Okay, great. Now notice how all of these lumbers coefficients are even which means we can factor it out to. So if we do that, let me delete this. Yes, on this. So we're who factor out the to the tennis, just turning to five. The's far is gonna train to to something with these four. Now we have a two indifferent are now based. Do on these two, they're gonna cancel out there like that. We just find or Italian banker Not within the city. Sanya factor to find a normal vector unit. Normal vector. So we're gonna take a distributor which is still going to be so she promised e is equal to 1/29 because that's a costing, so it doesn't change times out of pieces. So did you reveal to cause NT is negative to sign t on the derivative of this is negative two co sign t on that real five issues zero because five is a constant. Now I want to find a magnitude of this So the magnitude is going to be so the penalty. We're finding its my mind too. So we still have while always any night on then times the square root negative to negative two since number two some 70 square used to us for sine squared. Uh, 17. With this one, it's it. Instead, it's causing because the native two sons aged to sentence to a positive for we'll have us for. Oh, sorry. It's where as your times he was still cereal. So we just want to leave it like that. Okay, cool. So this is the magnitude. Now we're going to divide this by this. Now, notice how this is a fraction, right? So we're dividing this too. These two fractions are going to cancel out, right? So, what do you want? Anything like this? Just forget about everything that's over here on this is going to be on a normal back here. So everybody is equal to the care off. Negative to society. Home a negative too. Hold 70 Coma zero over. Now Notice how? Because of the same thing with the last time we just talked about the four on the science for plus persons 40 transit to one. So basically, I was gonna be left with on the coefficient. Is this brutal? For it's as we found out before this aspect of t plus causes Scrotie chances to one. Now, of course, Scooter for is just to those it's too. Now we continue to fight this even more because notice how we have a two right here. That's right here. This is just here So we contact our this year from each. So they say. So this is gonna be negative. One negative one. Don't tell it to right here. And we got to write. You write. So these two are gonna cancel that? I'm just on a recent, so it always more space. All right, let me circle this. Oh, of course. We don't need this line anymore as we're not dividing by any. So this is sort unit normal vector. Ah, now we could Who? The disease. The right answer By taking the magnitude of each one. Uh, this is what I mean. So basically, when you cut them by into the unit on your vector, it should be equal to the magnitude of the normal vector. They should be the same. So take the magnitude of this on the magnitude of this and see if they're the same. So first of my film, this is going to be one over this period of time. Nine. That's fair. Moves to cause anti times to cause entities just for call sign. It's where on this one is probably negritude. They get up to these positive for on scientists. Um, 70 East sign square see from five times five piece. Just 25. Okay, great. Now we can see the fighters were gonna back up before causing square plus course and square. I mean, close in support of people. Science, cruelty. It's just one. All right. So we have the course asperity. Well, science broke t before. That's what we d I forgot the teeth on 25. So this is off course. While I'm over this period of 29 we have four times one class 25 four plus 25. He's just 20 night like that for me. Put it like this. Now we have 1/39 times. Whatever. This fruit of 29 times this proved of 29. So this is going to cancel another. Well, that foot is one. Now, we're gonna do the same thing with enough tea, so we're gonna take its magnitude, which is negative. One sign. Lt those negatives wants. Lt is just sigh square. See on something with the course that's going to turn to course. Science parity on your hands here is just to so the distress into the spirit of wine that is off course, just one. So notice how this one give us? Why this one careless one? Which means that this is true. The monitor of tease people to the mind and which it means that this is right.

Given the curve T Square over two for minus 30 and one. We want to find what is the unit detained in factor T. What is the normal vector and and we will check that the normal an anti are one and the dot product. It's zero. So let's start by evaluating t. First, we have to find a derivative off are the derivative of our isn't the river off each coordinate so that there are is going to be the derivative of the first coordinate is to t over to which is teeth, and it deserved a second coordinate his minus three. The video of the third quarter insistence The scaler. It's going to be zero now. We find the absolute value, the derivative, the absolute value is a square root off each co ordinates square. So it's the square root off T square plus nine. Now, to find t, we must find the ratio between this to fact expressions that we found. So the ratio between these two is the ratio between these two expresses we found. So it is going to be t over skirt of T square plus nine and minus three over square root of T Square plus nine, and the other is zero over T Square plus nine, which is here. So we found T already. Okay, now we are going to find a normal vector. First we have to find a derivative of tea and then the absolute value of tea. This going should be really time soon, so let's start to have to do it. But enough teeth, derivative of tea is a derivative of this expression, so it's the derivative of each chord in it. The driven off the first coordinate. You can use the caution rule, and you find in the end says it's a bit time consuming. It's going to be 9/2 square plus nine, 2 to 3/2 for the second quarter, and you can also use the caution rule if you prefer, and it's going to be three t over. He squared, plus nine to the three of her, too. So now we can evaluate the absolute value or the length or the norm off. This expression is given by the square root off the first coordinate square plus second Cornet square plus 30 coordinates square. Let's do this. While the first coordinate square is going to be 81 over T Square plus nine cube. The second cornet square is lying T square over T squared plus nine cube the third coordinate square his zero now notice that 81 9 are multiples of mine, so we can put nine in evidence here To simplify the expressions, you're going to have mine times nine plus one T square and on the bottom we have t square plus nine cube. Notice that this city square plus nine one and this t scripless nine cube so we can simplify this and also noticed that nine is a perfect square so we can take the square root of it. This skirt of minus three and inside a square root. We are going to have one over T square last night square begin simplified this of in Florida or because the square root off a square often number taking the square he is going to be defaulted three over three square plus nine. Now that we have found that the absolute value off d T and D T, we can finally evaluate. And so let's do this If we divide this expression by this, we are going to have it and is equal to each coordinate will be divided by this number. So we're going to have nine over T Square plus nine 3/2 divided or multiplied if you prefer by t squared plus 9/3. Three t over T Square plus nine 3/2 is going to be divided by three over t squared plus nine, which is the same thing as multiplying by T Square plus line over three. That's coordinate zero so that no change so n is going to be mine divided by three is three one minus 3/2 is going to be negative. One of her too. So the first coordinate will be three over square root off T Square plus nine and the second coordinate. We can also cancel three and then we are going to have the second corner tee over square root of T square for nine. Now that we have found in and we have found also t, we can finally check if the alarm is actually one and the cross product is zero. To check that enormous one, we have to evaluate the norm off T and then arm off in dinar Mufti IHS taking the square root off all the cord and it's a square. This is a lot of computation, but we can do this. The first quarter net square is T Square over T Square over nine. The second quarter net square is nine over T Square over nine and noticed that since on the top of hefty square tonight, you know the bottom we have to square over. Nine. This going to be one. As for the length of N, we have that the first coordinates square is nine and the bottom is T Square, Pulis nine. The second coordinates squares T Square in the bottom, Misty square plus nine. And again we have T Square plus nine of the top T scripless line on the bottom, so the caution would be one now for the cross product. The cross product of two factors is first time's first plus second times, second plus third time's third Third time's third is already zero. Now we just have to check if this cross product is going to be zero. So let's do this. I'm going to use this space here to the problem, so T cross product and is going to be first times First, this is three teeth over. T scripless nine. Second time. Second, this is negative. Three teeth over T square last night. If we're them both, we have zero. So we found t we found in. We found the absolute values are equal to one and we found a day dot product is zero.


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