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Given r(t) = (tsint + cost)i + (tcost _ sint)j Find the curve's units tangent vector. Also find the length of the curve from t = V2 to t = 2...

Question

Given r(t) = (tsint + cost)i + (tcost _ sint)j Find the curve's units tangent vector. Also find the length of the curve from t = V2 to t = 2

Given r(t) = (tsint + cost)i + (tcost _ sint)j Find the curve's units tangent vector. Also find the length of the curve from t = V2 to t = 2



Answers

Find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$\mathbf{r}(t)=(2 \cos t) \mathbf{i}+(2 \sin t) \mathbf{j}+V 5 t \mathbf{k}$$

So we're asked to find the unit Tangent Vector of the following Parametric Curve R t. But we're given our prime if t is equal to e t t plus five sign of T plus two and CO sign O T plus two are my three components and we want to find the unit tinge in vector of our of tea at the point t equals zero. So how do you do this? Well, we know that the and again these are these air effect er valued functions. We know that the unit tangent factor T capital TFT is equal to the derivative of r of tea divided by And since the unit vector of of the unit Tangent vector, we divide by the norm of our prime of teeth. Now, if you look at the expression were given to find the norm of this of this vector would would not be the easiest task. But what? We're asked us to find the unit into Victor at a specific point. So what we really need to find here is t of zero, which would you need to find the derivative of r of zero over the norm of our prime at zero, which will be a much simpler task, since these will just be just numerical vectors. So factors with all the components are numbers. Okay, so what do we have here? So it's Write this down. We have our prime of zero is going to be e 20 plus five. Sign of zero plus two and co sign of zero to which we can see is just eat of the zeros. One So one plus 56 sign of 00 plus two is to co center. Zero is one plus two history. And now we can Now we can work this out because the norm of that is equal to the square root of each of these components squared six squared plus two squared plus three squared, which we get 36 plus four post nine, which is going to be 49. And the square 249 is sound. So then our answer t of zero our capital t 00 will be the victor six to and three where each of these is divided by the norm, which was seven. And that is the unit tangent Vector

You miss problem? We have a curve R t defined by a three course 90 till scientist. So the irritated off this curfew. Belief minus three slightly in the tool. CropScience When he caused this year old 10 jobs actor are crying. Zero he crossed the zero to So the union taken back at this phone you teach. Is there one the key close to I have the detention vector. Our prime off high half equals two minus 30 old. So the unique endure vector p off high half. We have B minus 10 and what he closed to minus pi. Half our prime off miners have equals 23 0th So the unit had your Magda tee off high at minus. I have It goes to one here.

In this question, we have a curve are key defined by three key and the piece worth its curative with respect to pee will be three to t. So when P causes zero, um, the tinges vector up, Frankie a friend zero will be 30 old. So the union tend your record. P zero is just 10 And when he closed to minus one, the tension vector our pride minus one Because doe three minus two and the units tend your record, he minus y was through three overweight, off 30 minus two over route off 13. And that's when he closed toe one. Um, the tension vector up from one because the three to the unit tender vector t one because the three over eat off 30 to overeat off 30.


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