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For Exercises $1-8$ you found $\mathbf{T}, \mathbf{N},$ and $\kappa$ in Section 13.4 (Exercises $9-16$ ). Find now $\mathbf{B}$ and $\tau$ for these space curves.
$$
\mathbf{r}(t)=(6 \sin 2 t) \mathbf{i}+(6 \cos 2 t) \mathbf{j}+5 t \mathbf{k}
$$

Buy a size loving. So by exercise. Excised loving in section The 18 points four. We have T. To be equal to because see minus sci fi provided by Square Root of two. Hi plus unless you have sci fi plus coast team Cost. He divided by square root of two G. Then in it's equal to minus cause T minus science C. Divided by square itself to spirit of two. Hi bloods minus sign. See let's cause see Divided by square root of two G. Okay so to find that by. No they buy normal Victoza which is B. To be called to the cost product of tea. And aim this is equal to I. You have because see minus Science C. Divided by Square Root of two. And this is minus cause C minus sign. See Divided by Square Root of two. Then for G. Companies you have sign see floods cause she Divided by Square Root of two. The N minus sign. See blood costea Divided by Square Root of two. Then I care components. It's the courts zoo. You have So here so the cost products of this give says so this gives us we have so of course the house cause squared see minus two coast sea 70 because T. Sign T. Plus sign squared C. Divided by two. Let's signed square T. Plus two Sci Fi. Cause T. Plus course course squid sea divided by two key. Okay so this is going to be equal to this week. You birds because see plaid sci fi. That is one to have one minus saying to T. Year for this divided by two plus stateside upswell gives me scientists scraped science grade D. Plus course. Great. See that is one plus sign suit see divided by two key. And this is equal to so this and that Give Me one. So this is equal to K. Also we know that we sue. RV. We it's record too. It's the power T. Cause see minus each the power etc. Sign. See hi plus he's the poet C. Sign. See plus eat the poet C. Cause see G. This in place that A. It's called soon. It's the power T minus sign. See minus cause see platz you have E. T. Out as well. So you have course T minus sign. See strain. C. Hi. Flags plus the J. Packs. We should give us E. C. The powered C. You have course see minus sign. T. Plus each the power T. You have sci fi 17 plus because see G. Which is equal to This is going to give us -2 U. To the power T. Sign. T. Hi less to eat to the poet C. Cause see G. So this implies that the let's see he's going to be equal to minus two. Is the power etc. You have course see blood sign. See I plus two. It's the power etc. You have mine. It's sci fi plus course see she therefore vehicles A have vehicles. E. Because A. V. Because A is going to be equal to you. Have I. So have E. To the forward. See of course, C minus science. C. Then minus to eat the power see saying. See then G. I have reached the power etc. Scientific plus. Ghostie sci fi plus, of course. See then I have suits each the powered C. Because see Then your key confidence you have 00 here. And this is going to be equal to shoots the power E. To the power to see key. So to find the magnet suit of vehicles here is going to be the square of that. So I have suit eats the power to it see screed. And this would give me four. Is the power for see. Okay, so if you find my toe shame. So the solution then is going to be cool too. Mm. Machine I have I have E. To the powered C. Yeah, of course. T minus sci fi. Then minus suit is the powered C. Science. See then minus two into the Poet. See you have cause see blood sci fi. Then the I. Components. So let's say I. G. I have each the Poet C. 70 plus Costea Science C. Plus. Because see and I have to E. C. The powered C. Because see Then two is the poet C minus sign. See class go see name. He. You have 000. Oh this is divided by These which is four to the four E. To the power for C. And this will give us zero at a finer Answing to that ocean. Then it's equal to zero.

By exercise 14. So by exercise 14 in section 18 points four we have C. To be equal to minus courts. See I legs sign, see G. Then in it's equal to sign, see I floods because C. G. Who is C. G. So this implies that by normal vets to be is C. Well it's mm which is equal to you have your eye, the eye components you have mine. It's cause see saying see the energy components. We have sci fi because see Then UK complainants you have observed zero. So they cross borders of this gives us mine. It's key. That's awesome. We have V. So my next key. You have the two vehicle to -3 cause squared. See sci fi I let's do we same squared see of course. See G. So this implies that A it's equal to the A. D. C. Equals C. D. Uber did see of minus three because squid sea sci fi I last d over DT of three. Science quake. See cause she G. So this implies that the air you ever did see it's going to be equal to de cuba D. T. Uh The Uber did see -3 calls squared T. Sign T. I plus D. Uber D. C. The Uber D. C. You have three. You have three science squared. See who's she G. Okay. So then to find a so shame we have. So this implies that you take the cross radars of minus three cores squared sine cosine squared T. So I see then you have D to D C minus three cause squared C. Saying see then we Uber D. C. Of D you ever did see of minus T. V course quaid C. C. T. Then you have three so squared C. Was see then the Uber D. C. Of three same square T. Cause see they have the Uber D. C. Of D. Uber D. C of three science squid. See this times square T. Cause T close T. Then you have zero for the K. Components And this is equal to zero. So this implies that I was so ashamed, is equal to zero.

Bye exercise 16 in section 18 points four. We have to see to be equal to one over square which of two slash see hi minus One over square root of two. Did you do Plus one Over Square Root of two. Mm. How this See then in its first two equal to Mhm. So right. Nine nights T so to find it by no more vector, which is going to be equal to see who's in. And this is how I To have one over square root of two. Right, see with these then this has that's then for I. J. Companies we have This. So when this one over square root of two, there is no J company. So this is zero. Thank you. Components I have this is key have one Square root of two here then key here is so I see. So my nets gosh, see and this is equal soon. All of us going itself. So here that's right. All right. 222 So by normal vector, it's equal to that. Also, we have the So we have the to be cool too. The Sash. See I minus Kash. She She Let's care. So this implies that E. It's equal so Bush sweet. I my Nice sash. See G. Which utilized that. So this implies that the A over D. C. Is going to be equal to slash I minus. Of course. See jean. So, our vehicles. See, we could say it's going to be equal to I. You have V. You have this fresh see a course see thingy, but it is minus Kash. See. Dane my leads. So sh see think a companies. We have zero. We had one here. So he had keys. So we had one named Sarah. This is equal to you have Sash. So sh t in the eye components plus, of course. See the J components. Last Kash squared close squared c minus sandwich squared C in the he components. And this is equal to this is equal soon, Sargsian. See Hi plus, of course. See she plus key so they might need to it. So the magnitude of vehicles E. Is going to be sash squared C less, of course squared C. Bloods when Mhm. So this implies that's for our socialism. I'm not so sure we have some cash suicide C course, see size, See, we have minus coche, See minus Samaje. See minus Kash. See You have 10. So Oh, divided by this. That's a cinch size squared C. Last Kash. Gosh, great. See that's one. This is equal to one divided by sabotaged squared C plus, of course Squared c plus one. And this is equal to negative one divided by to wash squared. See, it's our final results for the tour show, is this

So by exercise nine mm hmm. Exercise nine exercise nine in 16. In section 13 points four we have C. T. To be equal to Do we divided by five Cause T. Hi lights minus three Divided by five signs. See G. Bloods four divided by five key. And uh a. It's equal to minus sign. See I minus cause T. G. So what it means is that the binomial victor which is B. It's equal to the coast through that soft C. And N. So this is going to be equal to so you have your eye. So how I. So under eye you have three. Because because see three divided by five Coast Sea. Then you have my nets sign See then for jail gee you have- to be divided by five. Science see and of course minus of course see then for key we have four Divided by five years four divided by five. And this is zoo here. So this is going to be equal to This will give us four. So you have for the eye cos you have these so you have these times that in these times this so I have -4. I have this times this We should give me four. So I have four divided by five Cause T. I plus the J. Pads. So because of so you have plus minus plus so that becomes negative here. Mine it's so you have minus You have four divided by five sci fi. G. Then the last but you have three Divided by 5- Key. Also we know that V. It's equal to three course. See I plus minus three saying C. G. Plants for key. So this implies that our A. Yeah it's equal to minus three sci fi. Hi let's minus three. Cause see G gee that is for E. So D. A. D. C. Yeah the the A. D. C. Well give us if you differentiate sine you get cold so I still have -3 cause T. Hi Plus This will give us minus signs that have three signs see G. Okay then vehicles A. So this is fee this is a Sophie because E. He's going to give us you have your eye And I you have three three course T. Then he has minus three sign seen thingy for B. You have minus three science C. Then this is three CATT T -3 because T then for K. In the K. Direction you have four zoo. So the cost products of this is going to be equal to you have this. So I have this so this and then so two of so I have to off so of course see I minus two of sign. See G. My name's Nike 19. Thank you. So this implies that the magnitude of this. So therefore this implies that the magnitude of vehicles A. Is going to be this squared plus this grade plus that squared so I have so of course T squared plus minus so off. My name is too wolf sign. See squared plus mine it's night sweet. And this is going to be equal to 225. Therefore the tour scene. That's a shame is going to be equal to T. V. You have three cause T -3 Sci fi Then -3. Cause see You have -3 science C -3 Sign. See Then three sign. See In 4K. Because you have four zero. So divided by 225. And this is going to be equal to four points four times minutes. nine science squid. Sea. My name's nine cause squared C. Divided by 225 And this is going to be equal so you don't forget night can come out. So that gives us minus 36 Science squared plus course squared. It's one so you have divided by 225 and this is equal to minus for divided by 25. So that's ocean. It's equal to this. Then we had we are by no means a victor as this as well.


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0 1 1 ial 7a 1 1 L 2 1 2 1 li 4 [ L 1 2 1 1 2
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