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Let zi (3,-3,4} and (2,0,1}. Find the angle between the vectors the nearest hundredth radian: Note: Rounding 838.274 the nearest hundredth 838.27If Ilvll = Ilwlland...

Question

Let zi (3,-3,4} and (2,0,1}. Find the angle between the vectors the nearest hundredth radian: Note: Rounding 838.274 the nearest hundredth 838.27If Ilvll = Ilwlland the angle between- and wIs 0 = 3 find Izv 3wl.Find parametric equations for the line segment that starts at the point P(1,2,3) and ends at the point Q(6,_4,13).

Let zi (3,-3,4} and (2,0,1}. Find the angle between the vectors the nearest hundredth radian: Note: Rounding 838.274 the nearest hundredth 838.27 If Ilvll = Ilwll and the angle between- and wIs 0 = 3 find Izv 3wl. Find parametric equations for the line segment that starts at the point P(1,2,3) and ends at the point Q(6,_4,13).



Answers

Find the vectors $ T $, $ N $, and $ B $ at the given point.

$ r(t) = \langle \cos t, \sin t, \ln \cos t \rangle $ , $ (1,0,0) $

In this problem, we want to find the unit Tangent unit normal and by normal vectors, T n and B of the vector function Represented by are of T equals co sign of tea, Sign of tea and Natural Log Co sign of tea. And specifically, we want to find those vectors at 2.100 which we can figure out what t value this corresponds to by looking at the component functions. So we know that CO sign of T needs to equal one, uh, and that occurs AT T equals zero. All right. And we can check evaluate t equals zero at other component functions and see that it checks out. So first, let's find her unit Tangent Vector. All right. First, we need to take the first derivative of our our prime is equal. Teoh negative side of t her sign of tea and using the channel rule, we end up with one over co sign of tee times, negative sign of tea which we can simplify that last component function and right, it is negative tangent of t. And then let's take the magnitude of our prime, which is going to give us negative sign T Quantity squared, which is going to be signed squared of tea plus co sine squared of tea plus tangent squared of tea. And we know that sine squared of People's Co. Sanskrit of T is equal to one by the factory and identity. So we end up with one plus tangent squared of tea. And that is the same thing as seeking squared of tea or absolute value of sequent of tea, which, because we're looking AT T equals zero where seeking of zero is equal to one. We can drop her absolute value signs because we know that this is going to be positive for the value of tea that we're gonna be plugging in so we can then find our unit tangent vector t by looking at one over second of tea times the vector No sign of t. No sign of t a negative tangent. Lefty, remember that one over second is equal. Teoh co sign. So go ahead and rewrite this. Distributing one of her secret seeking of tear distributing co sign of t end up with negative sign of t co sign of tea co sine squared of tea and negative sign of tea as our third component function. So it's because if one oversee Kindt of tea is co sign of tea, it's times tangent, which is negative. Sign of tea over who sign of tea. So for that third component function, we're just gonna get negative sign of teeth. All right, that we want to evaluate this Oppa point t equals zero. So t zero Well, to, uh, negative sign of zero times co sign of zero sign of zero is going to be zero. So we're gonna have zero as the first Ah, component purse. I'm squared. Zero is going to need one and negative sign of 00 So here we have our unit Tangent Vector. All right, let's go ahead and find our unit Normal vector, which is represented by n All right. So first we want to take the first derivative of your unit tangent Vector t so t prime Milic it this form event right here. What we're gonna get is a negative co sine squared of tea plus sine squared of tea. What's applying the product rules? So that first component function, second component function. When we take the first derivative, we're gonna get negative two times sine of t co sign of tea The man third component function Take the derivative of that will get negative co sign of teeth and I'm taking the absolute value or sorry, the magnitude of t prime What we're going to get. I'm gonna go ahead and square our first component function here. So we're going to get sign of T to the fourth power minus two times sine squared of TV Times co sine squared of tea plus ho sign t to the fourth power plus second competitive function squared is gonna be four sine squared t care sine squared of tea They're component functions gonna be cose on squared of tea. All right, that's are like terms. This is going to give us sign of T to the fourth power used to terms we're gonna combine. I'm gonna write my co sign to the fourth power of tea first. Ah, the two that I just underline what's gonna give us to sine squared of tee times co sine squared of tea plus Carson squared t Oh, let's see, How could we simplify that a bit? So these right here I recognize that we can write as sine squared of t plus co sine squared of tea When that some is going to be squared, adding her sine squared of tea This by the Pythagorean identity is one. So we end up with square root of co sine squared t plus one, right. So then, for you, normal vector, it's going to be t prime over magnitude of t prime, which means we're gonna have one over the square root of co sine squared of tea, plus one times the vector negative coastline squared of T plus sine squared of tea. Negative two times sign FT co sign of tea. Negative curse sine of t. Right? And then we don't necessarily have to simplify that further cause we're just gonna go ahead and evaluate this AT T equals zero. So what we get is one over the square root co sign of zero is one. We have one squared plus one, which is two, and our denominator, our first component function, is going to give us negative one. We evaluated AT T equals zero second component function will give us the felt You 0/3 component function will give us negative one. So when we distribute the one of her route to you. Get negative one over route to zero. A negative one over route. You, right? That is our unit. Normal vector. Last but not least, we need to find our by normal doctor. We know that by a normal vector is equal to the cross product of the unit Tangent Vector and the unit Normal vector. Since we wanted evaluated at T equals one or T equals zero. Excuse me. We're gonna go ahead and across the T vectors that we found. So we're looking at 010 crossed with negative one over route to zero negative one ever route to. And when we take that cross product, what we end up with is negative. One of her squared of to zero positive one ever swear to to that is thereby normal vector.

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So you want to find these three vectors at this point over here? Uh, as you can see, we just really need to fill out these formulas. Let me throw these off side, and we'll start with the tangent unit. So we need to find the derivative of our So there's going to be our prime of t is equal to So this is just going to be well, we take the drove of each component wise would be to t and then to t squared and then one, and then we need to take the magnitude of this magnitude is going to be what we just square each of them and then add everything up. So it would be four t squared was for t to the fourth, uh, and then plus one square, which just one. And then we take the square root of this. So we plug that all in over here we get our unit tangent is going to be two t, uh, to t squared one and then all divided by the square root of four t squared plus or T to the fourth plus one. Yeah. Now, um, well, we need to figure out what is T in this case. And if we come up here and look well, our last position is just t. And that says it's just equal one. So we'll just come over here and plug in T 0 to 1 to get it at this point. Should be t of one. So that is going to be 2 to 1. All over would be four plus four plus one. So that would be eight plus one, which is nine. Route nine is three. So we just divide three into everything. So that would be two thirds. Two thirds, one third. So this is our unit Tangent at one now to find the normal what we need to take tea and take the derivative of it. So I'll first need to distribute this each of the positions in order for us to be able to take this derivative. So it's going to look a little messy now, unfortunately, so we'll have four t square plus four t to the fourth plus one all over two teeth. Then we have two t squared all over that magnitude again. And then one over this magnitude. Okay, so now we can attempt to take the derivative of this. And hopefully we don't mess that up because that is kind of involved. Especially let me just this up. Scoot it down here. So, the derivative, we're just gonna take it component wise for each of these. So we'd end up with t prime of t being equal to. So for the first one, we're going to use quotient rule. So it would be Whoa, Do you hide the derivative of two t is just too, um, minus high. So to t take the derivative of this. So actually, let me write that off on the side. So D by D t of square root of 40 square plus 14, 4 plus one. Um, so this would be to a one half power sort of be one half, and then the new power, since it's one half would become negative one half, which is the same thing is just one over that square rooted. So four t square plus four t to the fourth plus one. And now we can go ahead. And, um, what do we want to do? Take the drift on the inside? I don't know why it took me so long. to figure out, but it would be eight t and then was 16 t cubed. And so now we can get this to to counsel with that, that becomes a four that becomes an eight. So the derivative of this looks like it will be, um So I'll factor that four out So it would be or t was to t cubed all over the square root of Fourty Square plus 42 the fourth plus one, um, art, but now will plug that in over here. So this would be four t plus two t cubed all over the square root for t squared plus four t to the fourth plus one and then all over, um, the denominator squared. So that would just be four t squared plus four t to the fourth plus one. Okay, um, so let's see if we can simplify this at all before we go into that second position. Well, I could multiply the top and bottom to rationalize this, And in doing that, we would end up with so this would be squared. So it be four t squared plus for T to the fourth plus one. And there would be times two, and then we'd have minus two t. Or actually, if I distribute the two T, that would give us 18. Uh, and then that would be eight t squared, plus 16 t to the force and then all over. And then we'd have to multiply the denominator by that route. So now would be to the three halfs power. Mhm. Okay, uh, and then if I distribute this to 30 of us eight t squared plus 82 the fourth plus two, and then the negative would be minus 80 square in minus 16, T to the fourth, and then we can go ahead and combined are like terms here. So the T squares cancel out, and then we just have that, too. So it'll be too two minus eight t to the fourth, because we can combine those two t to the force, and, uh, I don't see anything else we can do. So then we just go ahead, and but what we have in the denominator again, So, yeah, that was pretty annoying to write out. And we only have to do that, um, two more times. So I'll actually do is take this and I'll do it off on the side again. So let's write that one up top here. So we have d by d t of two t squared all over the square root of fourty square plus or T to the fourth plus one. So again, we have to do quotient rule. And we already know what the Denon the derivative of the denominator, is going to be. So, um, we first strike this outside shall actually write it down here. 40 square plus war t to the fourth plus one. Then the derivative of the top is for tea, then minus those split. And the derivative of this is going to be for T plus two t cute all over the square Root for t Square plus four t to the fourth plus one all over what we have and the denominator. Uh, and so that would be for T Square plus four t to the fourth plus one. Okay, let me down Song. Um, So again, I'll go ahead and get rid of that square root like I did the last time. So then it's going to be or t square plus four t to the four plus one times for teeth and then minus. So that is going to be eight t squared. That would be eight t cute. Uh, plus 16 t to the fifth and then all over again. That denominator is now going to be, uh, squared. So it will be four t square plus. Or actually, uh, this is where Why did I after before? Hold on. Oh, because I was for something else. I don't know why I Rufete so this should have nothing down there because I should have squared it 40 square plus four t to the fourth plus one because this should have the same denominator as the other one. So this should be to the three house power. Now catch. Let me pick this up. Then we can scoot that down, so that should go here. Um, and then I don't think I have enough room, so I'll just do dot, dot, dot for that. And then we can go ahead and combined are like terms. So let's go ahead and distribute that fourty sewed beef 16 t cute and then plus 16 t to the fifth plus 40 and then minus 80 cute minus 16 t to the fifth. Um so these will cancel out, and then we can go ahead and combined those cubes. So that would be to my s a t to the fourth all over Fourty Square plus 42 4 plus one, raised to three half and then at 14, just days 14, and then it would be positive 80 cubes would be 80 cubed plus or t all over for Tea Square plus or T to the fourth plus one raise to the three halfs. Okay. And we're almost done because we just have to take the derivative of this last part here. So I guess I should just do this here because I don't see me actually coming and filling these in. Well, the only real difference between this is it's just reciprocated. Um, so we could go ahead and just use chain rules. So let me go ahead and do that down here. Mm hmm. My home. So, yeah, I just kind of scroll down a bit and write it out. So d by d. T of one and actually all right it as or t square plus or T to the fourth plus one. Raised to the negative one half power So now this is going to be so We use power organs would be negative one half and then that would just become negative. Three house to be fourty square plus four t to the fourth plus one to the negative three halfs. Then we take the derivative on the inside, which is going to be eight t Well, not 80 square, but just 18. And then plus 16 t cute. And then we go ahead and cancel this too with, uh, and that again. Then we can just go ahead and collect all those terms. It would be negative or T was 80 cubed all over now. Four t squared plus four t to the fourth plus one raised to the three house. Okay, so now let's go ahead and pick this up. Now we can put it right here, okay? And that is only the derivative of our unit tangent. That is joyful. And at this point, we can go ahead and find the magnitude of this. So if we get magnitude, we're just going to square all of these animal together, so this would be to minus 82 4th square. So if I square everything in the denominator is just gonna become what we have there. Cubed. So before t Square plus for T to the fourth plus one. Cute. Um, actually, they all have the same denominator, so I'm just going to write that once, so I don't have to write it out 1000 times. So mere race this mhm looks like it's flag in the little. So yeah, I'm just going to make this as one big fraction since again they'll have all the same denominator. So would be to my s 84th squared plus eight t ah, cubed plus 40 squared And then plus right, So I do something weird. Well, both of these should have been negative here, so that's a negative. Also, because the negative on the outside should have distributed to everything there. Uh, and then that would be negative for T minus 80 Cube squared, and then it would be all over again. The denominator squared, since I'd have to square each of those when I was adding all this up so it would be four t squared plus four t to the fourth plus one. Cute. And then we have to take the square root of all of this. Okay, Now we need to just expand everything there and actually, let me just get rid of this. We really don't need that anymore. And, um, if we expand all of this out in the numerator, um, this is just going to be perfect square and yeah, so it should be or minus 32 t to the fourth plus 64 t to the eighth and then plus 64 t to the sixth loss. So it would be 60 or t to the fourth, right? And then plus 16 t squared and then here, Actually, since I am squaring this, um, I can pull that negative out, and it would essentially just become positive, so we don't need to worry about that anymore. And then we would get 16. He squared, plus, um 32 64 t to the fourth, and then plus 64 t to the six. And then again, all over this denominator, meet the or plus one cubed. Um, actually, since the by normal vector, um, we don't necessarily need to get that exactly, because it doesn't depend on anything else. We could just go ahead and plug everything into this. Um, I guess we actually didn't need to find this exactly. I was thinking of how, like, we couldn't just use what we had before, um, to do this, which I mean, I wish I would have thought of that earlier, but, uh, yeah, let's actually get rid of all of this because we don't really need it. Um, and I'm just going to plug in one, and then we use that to find the magnitude. Uh, so yeah, sorry for attempting to go through that, but Yeah, I just kind of thought about how we could do this. Um, because again, I, for some reason, thought we needed that. Uh, yeah. So let's plug the sense that would be just to minus eight over and then be four plus four plus one raise to the three halves over here. It would be eight plus four over. Well, that would just be nine raised to the three halfs. And then over here, it's going to be negative or minus eight over nine to the three halfs. This should be TF one. So that would give negative six over. Uh, well, nine to the three Halfs should be 27 because it would be, like three cubed. Uh, this is going to be 12/27 then this is going to be negative. 12/27. All right? And now this is something that I would much rather find the magnitude of. And so now we can square each of these components to be negative. Six squared plus 12 squared, plus negative. 12 squared. And then this is all over. 27 squared, and we square root the whole thing. So this is, uh, 36 plus 1, 44 plus 1 44. And, um, so 3. 20 for over 27 squared. And if we square both of those 3 24 is actually 18. Would be 18/27. Well, okay, So our normal sector in T again was supposed to be t prime A t over t prime of T magnitude. But since we just want this at one, just go ahead and plug in one. So let's come up here, so I'm just going to pick this up. I don't have to write it out again. So this is going to be this. And then we divide by 18/27. Okay, um, so this would just become negative. 6/18. 12/18. And negative. 12/18. Um, and then I can just divide the top bottom of all these by six. So that is going to be negative. One third. Two third, negative two thirds. And so that's in a one. And lastly. So actually, let me come up here and hold this down so we can try to collect everything together. It's all need this or the next part. So let's come up here. So to get the last one we need to do our by normal, we need to do the unit tangent with the normal cross product. So, um, we want b of t. And this is going to be almost order again. Uh, t then in. Okay, So it would be t of t crossed within of t. So to do this cross product, we have i, j and K. And then we put t first. So it would be two thirds. Two thirds, one third, and then we would do negative one third, Two third negative two thirds. And actually, this is going to be at e one, all right. And Now we can go ahead and just take our determinant. However, I normally do, um, this method for it. So I rewrite the first two, the V i two thirds negative. One third j two thirds. Two thirds. So I do my downs here. First I multiply those together, Adama. So it would be, um, two thirds times negative two thirds, I plus one third times negative. One third jay, and then plus two thirds times two thirds. Okay. And then I subtract going up the column. So let me scoot the silver. So minus. So first, we'll just write negative one third, two thirds k so you can see how I'm going up the column, then Plus two thirds. One third, uh, I So two thirds, one third eye. And then it would be negative. Two thirds. Two thirds j. And that's going up like that. Uh, Now we just multiply. All these, uh, combined are like terms. Let's see, um, this here would be for or negative four nights, I That's going to be minus minus 1/9 J and then plus 4/9 K. And then this would be minus so two nights negative two nights. Okay, um, then plus two nights I and then minus four nights. J uh, actually, let me just go ahead and make sure I didn't do anything weird with this first. Yeah, everything looks fine. Uh, I think I wrote everything down. Right now, we just go ahead and add everything up. So distributing this negative, so that would be positive. This becomes negative. That becomes positive. Um, so eyes are going to become negative. 69 the chaise are going to become 3/9 and then the K's are going to become 69 which we can go ahead and simplify it down to negative two thirds, one third and two thirds. And so then this is going to be our by normal at one. Mm. Um, so let's put all of these in one place now. Yeah, Mhm. Because we've been doing this for a while. We actually even hit the page length. We have just how much work we had to do. Yeah. So these are going to be those three vectors that they asked for Just a ton of work. Um, wish it wasn't as difficult of a process, but unfortunately, that's just kind of what we have to do.


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