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Parallel planes Find an equation of the plane parallel to the plane Q passing through the point $P_{0}$$Q: 4 x+3 y-2 z=12 ; P_{0}(1,-1,3)$...

Question

Parallel planes Find an equation of the plane parallel to the plane Q passing through the point $P_{0}$$Q: 4 x+3 y-2 z=12 ; P_{0}(1,-1,3)$

Parallel planes Find an equation of the plane parallel to the plane Q passing through the point $P_{0}$ $Q: 4 x+3 y-2 z=12 ; P_{0}(1,-1,3)$



Answers

Parallel planes Find an equation of the plane parallel to the plane Q passing through the point $P_{0}$ $Q: 4 x+3 y-2 z=12 ; P_{0}(1,-1,3)$

Hello. So the question is taken from victor's and geometrical space. And the question is find the equation of plain. That plane contains the line. It means the plane passes to the same point and from which the line passes. And uh it's popular to the plane five X plus two Y plus that is equal to one. Okay So let me right here that uh that 1 2nd mhm equation oh clean passes to slowly contains the line. Yeah X. is equal to one plus d. Bye. Sure is equal to minus T. And that is equal to 4 -3 T. And he's Caroline buddha clean five x plus two. Y Plus they're difficult to one can be who are you Dennis Since both plane passes to the same point as the line passes and the line passes to so that by taking the T value equals. So we get equation of Linus X -1 is equal to Y -2/-1 And is equal to that minus for -2. Which is the equation of line. So since line and plane passes to the same point so the point from which the plane passes to is 1 to 4. So that equation will be five in two x minus four since it is regular to the given line. So that equation will be five in two X minus four Plus two into Y -2. You can also evaluate by taking it a like since it is particular to the plate. So that will be oh five five X plus two. Y plus said plus lambda is equal to zero since the plane passes to 1 to 4. So from here lambda will be equal to five into 1 plus two into 2. Plus that into for the physical too. Sorry The sources that is not five into 1 Plus two into 2 plus On place of that there will be four is equal to minus of lenders. Or from here lambda will be equal to five plus four plus four. That will be equal to minus off there. Or you can say you can take a minus sign here. Okay? So from here and I was able to 13. So let me substitute this value. And the equation where you get five X plus two Y plus debt minus 13 is equal to zero, which eventually gives the equation of plane is five X plus two. Y plus Z is equal to 13 because the plane and that the human line passes to the same point and it is popular to the given place. So I hope this clears your doubt and thank you

Did you find it? Question in the plan, Sister is going through the 0.1 months 13 and in its parallel, John on the plan three x plus. Why? Plus say in court, you seven, you noticed them with Mexican information. We can find a normal that the other plan we're looking for. Jenna Coefficient in front. Under explains he hear their 403 +11 Therefore, the question of the plan, it would be Hey, we have a three and then actually exerted Could you know one here? Why, it will be one and then we have Why is there Rico Jude minus one? So her plus one here. And so you have the plus one and he's a reported three. So the ministry cultures there are. And now we should simply find again in three x Ministry. Bless when I rest, one less the ministry e coaches era and one most. I'm gonna three x plus y plus c. They were ICO Jew Here will be five

The Plan X Plus two y Ministries. They go to one and identify the question of planned. And but this plan's off a show that should be the same. No more factor will be the experts. Teoh man ist she was They could use some constantly here and from the problem. 53. Won't you have this thing? This plan here and we condition with B B B once impeach you. So we want to have the distribution to be one and a peach. You in a coaching that you and record in the papers Question 53 this distant in Cochin Uh, de miners Ah, one absolute value divided by the square root off the normal vector This number better will be one less for less far. So get Nico Jew the demons one on bended running by three. So from here we can see that the afternoon d minus one with Nico to the six. And from here we should get the D minus one include your blessing minus six. Therefore we have being Whitney go to 1% minus six on again a t and we caught you. Everyone used the blessed with under seven and we use a minus. We get a minus five. Therefore there a question on the plan we're looking for have two of them. The 1st 1 will be expressed to why man ist Uzi. And in will be Ah, Echo Judah seven and Blessed Teoh Man ist Uzi include U minus five.

So here we want to find playing that passes through these two given points and is perpendicular to the given plane. Remember what we need for a plane as a normal vector or equivalently two, just two vectors in the plane from two vectors in the plane. We can take their cross product to produce a normal vector so we can get one vector just by the vector that goes from P one to pee, too, which we get just by subtracting the coordinates. Okay. And we need another vector. So we have one other piece of information we need to use. So the normal vector for this plane, maybe I'll call it V could just be right off. Is four minus one too. Okay. And so saying that our planes Let's say, here's my plane that's given to me, Calientes of my normal vector. The is like that. And we're saying that the plane we want is perpendicular to this one. So if you try and draw a picture of a perpendicular plane would look something like this. Okay. And what we noticed about it is that that this, um normal vector of the perpendicular plane actually lies inside the plane we want. So this is another vector in our plane. And so now we can find a normal vector by crossing these two. Okay, When we do this, we get negative too. Negative. 12. Negative, too. And again, since we don't care about the direction, maybe we can simplify this by dividing by Negative too. Sorry. I said we don't care about the direction. Actually, the direction is the only thing we care about. I meant to say we don't care about the magnitude. So that's why I can reschedule this. That arguably looks nicer than this one. Okay. And now we have enough to write down the equation for the plane. So the equation for the plane we get a zero is equal to the normal vector dotted with X Y Z minus one of the points in the plane. So I'll do use p one. And how does this simplify? We get X minus one plus six Y minus 12 plus Z minus three. And so we can write. The equation for the plane is expose six. Why close? Ze is equal to 16


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