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Question 6: Fiud thc Tangend vector; the Normal vector anc thc Curvature of the curve F() = V2sin /i V2 sin tj 2 cos (k....

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Question 6: Fiud thc Tangend vector; the Normal vector anc thc Curvature of the curve F() = V2sin /i V2 sin tj 2 cos (k.

Question 6: Fiud thc Tangend vector; the Normal vector anc thc Curvature of the curve F() = V2sin /i V2 sin tj 2 cos (k.



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Find the curvature $\kappa,$ the unit tangent vector $\mathbf{T}$ the unit normal vector $\mathbf{N}$, and the binormal vector $\mathbf{B}$ at $t=t_{1}$. $$\mathbf{r}(t)=e^{7 t} \cos 2 t \mathbf{i}+e^{7 t} \sin 2 t \mathbf{j}+e^{7 t} \mathbf{k} ; t_{1}=\pi / 3$$

We're asked to find the curvature that, you know, tanja vector? The vector and the by normal vector, which is also a unit vector at um that Tina equals two. Uh Yeah, so that's our certain point. I guess they call it the one called tina. Um So our curve is then T squared X equals two squared over two. Why it goes to Z equals T. T over three. So this is basically just an exercise and about taking a bunch of derivatives and doing a bunch of algebra. Careful. So first derivative of this with respect to time gives us this and that's a tangent vector, but it's not a unit vector. So we can find its magnitude and divide by its magnitude to get the unit tangent vector which I use a hat. And you don't need did you know to donate to note Anyway, so that's the unit tension vector. This divided by this, so it's a unit, you know, every week. Um We can then find the curvature by taking the magnitude of the derivative of this factor divided by this factor. And that winds up after some sense location, winds up with this value here. We can then find the normal vector by taking the derivative of this factor and normalizing it and that gives us this ugly thing here and then the by normal we can find by taking the cross product of tea. And so that one's just some vector calculus or vector algebra. So we take that cross product. And so you know, it's just all these normalizing factors are what basically makes everything ugly. You know, these things aren't that bad, right? But this the normalization factors that get everything ugly. So they asked us to say, well plugging teeth was too, So our curvature winds up being this or 0.0 Um 597. A tangent vector unit tangent vector at that point. Is is just this given numerically unit normal vector is this? And numerically and a unit by normal vector is this or in America is this? So um again, uh these are hard to visualize. You could actually, you know, if you make a plot and plot these rectors on there, you can see that. In fact they should look right. Um but they didn't ask us to do that and that would be kind of a just a check. But other than that is basically just trying to make sure you do your take the derivatives carefully and do the algebra carefully and learning how to use a computer to do all this calculus for you. Um Well, come in extremely handy because it won't make a mistake unless you obviously give it give it incorrect import or but yeah, learn to use a you learn to use mathematics or some other computer algebra system. Um That will do this calculus for you and it will make your life a whole lot easier

Given the curve actually cause either the minus T. Why it was either minus two T. By equals either of the two T. And z equals square to times square two times T. Um It was just kind of an oddball curve. I don't know what um what it represents. Maybe should have plotted it, take a look at it. But yeah, it looks like kind of uh yeah, I don't even know what you've caught. Um But we can still find not tangents normals and by normal curvature and that stuff, so we take the derivative and divide by the magnitude and get the unit tangent factor, take the derivative of this and divide by its magnitude. And we find the unit normal vector. And then we can take the cross product to find the unit by normal factor. The curvature. We can find again giving these two vectors, it's pretty easy to find. And we get this expression here. So they asked us to look at the point T equals zero. So at least it's a convenient point to look at. And in that case we get that the curvature is 1/2 times square 22 or zero point 354 The tangent vector is minus one half, I plus one half J plus one over square to to. Okay. No, that's the unit Tan director. Um Then the unit normal vector is one over the square to to eyes. Uh I plus J. So that's just in the um in this case there's no K component. So it's in the xy plane, it's just kind of a 45 degree vector, 45 Degree Angle Factor. And then the by normal by normal is minus what happened, The guy um Plus 1/2 minus um one over square to two K. So that's let's see here, yeah. Um Is that right? Let's see here. This winds up being 1/2. This gives us a -1, this is one and then this winds up so that's right. Uh Let's see here. Yeah. Yeah. Go again. Hard to really. Um Look at let's say you think, you know these things reasonable? I guess we could plot this curve around T equals zero and and take a look at, you know what the curvature is and stuff look like there and make sure that these vectors are kind of pointing in the right directions. And again look tangent, looked normal and look by North. Yeah.

Given this curve here, it's in a plane and the XZ plane. So it's X. Equals the Carson cube of T. And Z equals sign. Keep the tea. And if we plot that, this is what it looks like Right kind of this four pointed star thing. And we can figure out the uh the normal that we can again do all this stuff. Take the derivatives, calculate everything, um take derivative of this divided by its magnitude. Um You know then calculate this and then they cross product and this one clearly and this one that you can basically tell that if you get this, you know, should be out of out of this plane. So it should be in the Y direction. Because why is this case out of plain direct? We have X, Y and Z. Now so that we should clearly see that this. It's a good check. Um But then they gave us this point T. was pi over two and that's right here. So it's right at this cusp. And so if you look at these you can expand this out and simplify and right in lots of different forms, but at pi over to everything kind of breaks down falls apart, curvature again Falls is undefined, you know um basically infinite because you know we're at a cusp here at you know Because this has gone to zero, this is going to 02, but you can take a limit to see if we can from um And then these guys, you know when tea was pie over to, you know this is this is zero and what is this a sign of Yeah, basically the SGN of zero, You know, is that 0? What? And then this one here this is one, but then this is also you know the the sign of wait a minute, This was paid over two. So this was yeah well this is this is the This is zero. So clearly, okay, so the this is zero and this is this is one and this is one. So basically this is clearly zero, but here we have the sign STN of zero. What is that? This is 1 1 times. Is that zero or what again? So basically it's it's kind of either this way or this way we don't know. Okay, so it's kind of plus or minus K. Wait a minute, tangent. Tangent is plus or minus K. This way, this is the tangent may draw the normal, the normal, but you can see the same, the same problem happens. So it's either plus or minus. So I don't know, basically it's not really defined exactly at that point because it's a cusp this is clearly defined because it's always pointing out of the playing here. So it's kind of, I don't know, it's kind of a tricky question. Um so whether we say it's this way or this way, which way it is, because it's at this corner point that it's really these things aren't well defined at that at that point.

In case you have already liquid to sign of tea and good for tea and co sign of tea were asked if I curvature So it's first derivative. You get too close, Entity. They're gonna four and the good of society. I think that second derivative, you get your insanity zero and they're going to see a chorus entity. Now, let's take the cross product of our across our crime cross our, uh, double friend. So we get that people too, by J J. To call. Sign of tea. Negative for you to sign of tea. They have to sign it. See negatives to sign of tea. And Okay, Simplifies to eight. Co sign a C I plus for J minus a sign of teeth. Okay. Now, taking the magnitude of our prime cross our double prime you got. This is equal to four square root of five. We also need the magnitude of his art crime. What is that? It's his equal to group of to co sign squared T squared plus 16 plus to sign J R. 70 toward that gives me to screw to five are curvature is the magnitude of our prime cross Our double time So that gives me for primitive slide over Jews Group of Five to the power of three. This gives me one over time.


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