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(5 points) Parameterize the part of the plane 6r + 3y+2 = 12 that lies in the first octant: r(s,t) = (6s,3t,12 - s - t) with c [0,22] and e [0. 4 - 2s]. r(s,t) = (s...

Question

(5 points) Parameterize the part of the plane 6r + 3y+2 = 12 that lies in the first octant: r(s,t) = (6s,3t,12 - s - t) with c [0,22] and e [0. 4 - 2s]. r(s,t) = (s.6,12 _ 6s 30) with e [0, 2. AHd (€ [0,1 2s ' r(s,t) = (8.t,12 - $ -t) with $ € [0,2| and e [0,4. D. r(s,t) = (8,4,12 - 6s 3t) , with s € [0,2], and t e (0,4}: None of the above

(5 points) Parameterize the part of the plane 6r + 3y+2 = 12 that lies in the first octant: r(s,t) = (6s,3t,12 - s - t) with c [0,22] and e [0. 4 - 2s]. r(s,t) = (s.6,12 _ 6s 30) with e [0, 2. AHd (€ [0,1 2s ' r(s,t) = (8.t,12 - $ -t) with $ € [0,2| and e [0,4. D. r(s,t) = (8,4,12 - 6s 3t) , with s € [0,2], and t e (0,4}: None of the above



Answers

Match the planes in (a)-(d) with one or more of the descriptions in (I)-(IV). No reasons are needed. (a) $\quad 3 x-y+z=0$ (b) $4 x+y+2 z-5=0$ (c) $x+y=5$ (d) $\quad x=5$ I Goes through the origin. II Has a normal vector parallel to the $x y$ -plane. III Goes through the point (0,5,0) IV Has a normal vector whose dot products with $\vec{i}, \vec{j}$ $\vec{k}$ are all positive.

Hello, everyone. The question is that we have to write the equation off plane when three points on a plane or given we know the general equation off plane that we can right by if we have a point on a plane and a victor normal to the plane. So here, in this case, three points on a plane are given so we can find a vector perpendicular to the plane by drink to Victor's joining. Sorry, but trying to Victor's joining these two points and then taking the cross product get a normal vector to the plane. So first of all, I just look at the points that air on the plane that is okay. Three comer wonder three coma, negative life, and then cue, that is for common. Two or three comma, negative three and then are that is two comas. Miracle no one. Now let us find to Victor's on the plane. That is Pete, you. That is going to be one comma. Assert one by three. Come on. Two. And then there is br that is equal to negative to coma minus two. But three. Uh, full. No, we can take the cross product of these two victors. That is I into he'd write three minus eight g. No, this is considered this as a point on a plane and this victor as normal to the plane now we will use the general equation off. Mean that is off this from into X minus X not lesbian do. Why minus Why not? Let's see and do said mine, is it not? You want to zero Now we will plug in all these values where a, B and C are the components off normal vector and ex not why not? And that not not at the coordinates of the point given on the plane. So let us plug in values for this case. It is going to be X minus three y equal to do. This is the equation off plane in this case, and that's the answer.

Were given a surface integral and were asked to evaluate it. This is the surface integral across the surface s of X z. Where s is the part of the plane to X plus two y plus Z equals four that lies in the first often. Well, rephrasing this means that s is the part of the plane. Z equals four minus two x minus two y which is over region of integration de which is going to be the set of all pairs x y such that well, this plane has an X intercept of X equals two and A Y intercept white was too. So we have the X Cumbie at most two being the first doctrines that also has to be greater than or equal to zero. And if we restrict X to these values, then we have that Why has to lie between zero being the first options and solving our equation By putting in Z in terms of X, we have why must be less than or equal to X minus two. Yeah. Sorry. Not x minus two to minus X and therefore the surface integral of X Z over the surface s This is the double Integral poverty domain of integration. B of x times z in terms of x and y, which is four minus two x minus two y times The square root of the partial derivative of Z with respect to X, which is negative. Two squared plus the partial derivative of See What? Expecto? Why? Which is negative. Two. Squared plus one. Yeah, and writing. This isn't iterated integral. This is equal to well, the evaluates Get square root toe nine. It's free speaking factor out of three times the integral from Mexico zero to integral from white. Go zero d y equals two minus X of commanders. We have four times X minus two X squared, minus two x y b y the x Taking the anti derivative with respect to why this is three times integral from 0 to 2 of four x y minus two x squared y minus X y squared from 0 to 96. Yes, and evaluating their plugging in for why we get three times the integral from zero to or X. All right in minus X minus two X squared minus two minus X minus X times two minus X squared. Yes, and multiplying all this out and canceling on adding, like terms. You get three times the integral from Mexico, 0 to 2. We have an execute minus four x squared and plus four x The new constant term. Yes, taking it the derivative. This is three times 1/4 extra four minus four thirds sq plus two X squared from 0 to 2 and evaluating this is three times four minus 32 theories, plus he in simplifying. This is equal to four.

Nice in this Berlin your give entry points P history over to coma or common native to you, that is netted. One. How come on Tau zero And our that is never if one how come a zero comma two on what we want is an equation of the plane that passes through the points P Q and R So let's get started. So first of all, let's Quincy there this, uh, vector the better be cume onda vector p. R. So we think about the cross product off the factors that start this new factor and waiting about these, um, this new vector Deneuve actress perpendicular to both the vector P Q and A vector Pierre under, for it is perpendicular to the plane to P. Q and R. So if we remember Doc Cross product is defined in the next way. Just as I wrote before. So for the windows were more that we can get that the components off these new bedroom. So first of all, we just need to find out back the components off pick Yu NPR individually. So let's do that soapy cume. It is just tree over to minus and never early one house that is just for house. That he's just on that is just too. And we have four minus two. That is just too on narrative to minus zero is just a young girl to my No. Zero is just never victim. So those are the components off pick you now, PR. We have, uh, tree over to my nurse nearly 1/2 thatis again too. Then we have four minus zero that he's just wore on. We have narrated to minors to the majors and nearly poor. So those were the components off this factor. And now that we have the components of those two factors, we can get that, um, these the cross product off the back trippy que onda better pr. So let's just do that. Um, yeah, so following the formula, we have a two times be tree that he's Neri bait and we have my nose A tree. Me too. So that is just, um a tree is narrated to be, too is for so that he's netted a woman. We have a tree, b B one so that use nearly four minus a one between. So that is never a I. Finally, we have Hey, one thing's be too. That is a times Ah, do they? Minus 382 things be want bodies for so we can simplify those components. So we know that netted a a minus native ate that He's just a zero then 94 minus narrative a minus negative eight home that he's just for, um for minus eight is judged for so yeah, but now we have the components over these normal back her on DDE. Yeah, it is really important to how these neural better so we can get out who's, you know, off the plane. So let's just, uh, remember, let's continue on his page. Uh, let me, uh it raises. I'm really sorry. Yeah. So it is important to mention that if we have a point, let's say ex not. Why not? See? Not. And we have a normal vector, ABC then that plane that contains these points on these normal vector is described in the following way. Uh, eight times x minus X not blessed be. Thanks. Bye. Minus Why not Pelosi? Tanzi may know Xena on that is equal to zero, so we can't We can't take this into account on estar finding the equation off the plane so we can use of reference his point. The 00.0.3 over a tomb for my four common narrative too. There's just righted. Itwas tree over too. Well, not for culminated two on. We can't just rewrite the components off or back. Drew. And those components were zero coma for coma or and we conserve. Stick to those valleys e to our formula. So yeah, let's do that. It will be zero times X minus tree over too. Blows four kinds. Why minus four? Loves four diamonds e minus. Negative doom. And that is just it. Well, do zero. Yeah, I forgot. This parenthesis is so this is this will be just 20 on days. Will be for why? I know 16 up close for Z. Let me just re cried. Theis, please. Yeah, plus for Z Um yeah, No, anything negative is reciprocity of So we'll have. We will have z plus. Do so for if we applied distribution, we will have thes plus eight. So four times two is just ate so well have we will have a and that is just equal to zero. So again we can isolate the wise and disease. There's just Rick Rightist for awhile. I pose for sea and we can add 16 in both size and sub strapped a dreamboat site. We will get 16 mines, eh? And that is just a I'm Finally, we're gonna divide everything by four. So we will have ah, for at why I'm sorry. Blust Z on a divided by four is just too. Yeah, This is a red equation off the plane is why plus c equals tomb. Mmm. So, yeah, I does he answer why? Plus Z is equal to do. That's the question off the plane, so thank you for

Cannot problem twelve because why plus three equals five. Now, this problem is a little bit different. And two problems we did before, because here we have, um, no access here. Which means we don't have X is now X intercept in this function. So this plane's going to be parallel two x axis. So let's just frothy. Perhaps his first. Since we have no x intercept, we can't find a why intercept is gonna be We're planning X. It was there. Why was there? Another is just to be here. So zero So Winesap is just going to be Why coordinate equals five card zero zero zero five zero. Now we're plugging zero five zero here and all things considered intercept plugging ex wife was zero. So why was there? Here it is just five toe There. There, five days R, stay in the car to intercept zero zero five. Now we just need to collect this one. And since the play, it is parallel to X axis. So it's going to be something like that. Wait goes this way and there's no X intercept and Tencent floor problem


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