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Equation of the plane through three points $A, B$ and $C$ with position vectors $-6 mathbf{i}+3 mathbf{j}+2 mathbf{k}, 3 mathbf{i}-2 mathbf{j}+4 mathbf{k}$ and$5 ma...

Question

Equation of the plane through three points $A, B$ and $C$ with position vectors $-6 mathbf{i}+3 mathbf{j}+2 mathbf{k}, 3 mathbf{i}-2 mathbf{j}+4 mathbf{k}$ and$5 mathbf{i}+7 mathbf{j}+3 mathbf{k}$ is equal to(a) $mathbf{r} cdot(mathbf{i}-mathbf{j}+7 mathbf{k})+23=0$(b) $mathbf{r}$. $(mathbf{i}+mathbf{j}+7 mathbf{k})=23$(c) $mathbf{r} .(mathbf{i}+mathbf{j}-7 mathbf{k})+23=0$(d) $mathbf{r} .(mathbf{i}-mathbf{j}-7 mathbf{k})=23$

Equation of the plane through three points $A, B$ and $C$ with position vectors $-6 mathbf{i}+3 mathbf{j}+2 mathbf{k}, 3 mathbf{i}-2 mathbf{j}+4 mathbf{k}$ and $5 mathbf{i}+7 mathbf{j}+3 mathbf{k}$ is equal to (a) $mathbf{r} cdot(mathbf{i}-mathbf{j}+7 mathbf{k})+23=0$ (b) $mathbf{r}$. $(mathbf{i}+mathbf{j}+7 mathbf{k})=23$ (c) $mathbf{r} .(mathbf{i}+mathbf{j}-7 mathbf{k})+23=0$ (d) $mathbf{r} .(mathbf{i}-mathbf{j}-7 mathbf{k})=23$



Answers

Find an equation of the plane.

The plane through the point $ (5, 3, 5) $ and with normal vector $ 2i + j - k $

Hello. So the question is taken home vectors and geometry of the space where we hear we I need to find the equation of plain that plane passing two minus 11 by two and three. And normal vector I plus four plus case. So given the point is minus one. One by two and three. And it's normal victories. Mhm. I plus four G plus K. Okay so we know that uh equation of plane. Sure. Yeah passing to what? Passing two X zero Y zero and there's zero. And uh it is normal to victor Ai plus B. J plus see case A into -X0. Let's be into Y -Y0 plus C. Into k minus K. Zero is equal to zero using the same form here. So we get What explains went into one plus Fall into Y -1 x two Plus She's three in 2. So he went into 30 -3 that will visit. So if you're dead Zed minus that zero that is equal to zero. The value of C. Is one minus that. Zero is three. That is equal to zero. So that equation will be X plus one plus four. Y minus two plus zero minus three is equal to zero. So from here the value of equation is expressed for white list said is equal to minus five plus one is Plus 4. 1. That I can say which is the required the creation of plain that passing to that point and it's normal to their director. So hope this close your doubt and thank

Hello. So the ocean is taken from vectors and geometry of the space. And uh we have to find the equation of plane which passes to the 0.11 by two and one by three. And paddle to the plain. X plus Y. Plus that is equal to zero. So equation bulls clean passing to 1 1 x two and 1 x three. And yes Caroline. Mhm. To the plane X. Plus Why? Plant that is equal to zero. Can be. Have you done is okay? Yeah. Mhm. Taking X plus Y. Plus zero plus. Lambda is equal to zero. Substituting the value of experience that we get one plus one by two plus one by three is equal to minus of salamander. Celinda will be so that will be 3 to 6. So six plus three plus 2/6 of -10 Lame. That will be equal to minus 11 by six. Substituting this value in equation one we get six X. By taking the calcium of you can see by ghost multiplication plus six five plus six dead is equal to 11. Which is the required equation of plain. They're passing to 11 by 21 by three. And it's no uh pallor to the plane explains why because they're difficult. So hope this clears your doubt and thank you

According to the question, we have to find the equation of the plane that passes through the .314 and contains the line of intersection of the plains. Given by the equation number one express two, Y plus trees that equal to one. And equation two is two, X minus Y plus Z is equal to minus three. So in order to find the equation of the plane according to the given equation and point given in the question. So firstly they have given a point, let's say the point is E. And the coordinates are given us 31 fools. And in order to find the question of the plane we first have to find a normal victor And using these points we can find the equation of the plane. So in order to find the normal victim to the plane we have to find another. Mhm two points in the plane using the equation one and two. Therefore if X equal to zero then Equation one and 2 becomes do Y plus tris ID Equal to one and why minus way less? That is equal to -3. And by solving the simultaneous equation continues the two variables Y and there we can find out available. So if we multiply equation do with two then It will become -2. Y Plus two, said is equal to In place of minus treat will become -6. So solving mm The equations we can get by adding distributions, we can get that pi Z is equal to minus five, therefore Z is equal to minus one and in order to find out the value of why we can put the value of set in any of these equations that is minus Y plus said equal to minus three. Immigration to if you put the value of $0.00 -1 then it will become Why is equal to 3 -1, that is equal to two. Therefore we got another points. Let us say it be which has coordinates zero, two and minus one. Yes. Yeah. Yeah. And in order to find another point in the plane, you have to consider the value of the Is equal to zero. Therefore equation one and 2 becomes X plus to way equal to one and two weeks -Y Equal to -3. So in order to equate this to value and catholic the values of the variables multiplying equation two by two. We get This way efficient as four. This is too And this is six. Therefore adding these two equations. We can get the value of X, that is five X equal to minus five. Therefore The value of x equal to -1. And putting the value of X. In any of the equations we can get the value of Y. That is x minus y two, X minus Y Is equal to -3. Therefore go into minus one minus white. Well to minus three and therefore the value of why is equal to minus two plus three, that is equal to one. Therefore we got the coordinates of another point. Let's suppose the point. Because they see these coordinates are minus one one and zero. Therefore The three points in the plane. Oh point E 314 point b 0 to -1 buoyancy -110. Therefore in order to find the normal vector to the plane we have to find the two lines from these points that is line A B vector is equal to 0 -3. I gap Plus 2 -1. Jacob Plus -1 -4 K Cab. This is equal to -3 icap plus Jacob -5 K cups and this is equal to victor once opposed. And another line vector that is a C vector is equal to -1 -3, icap plus 1 -1. Jacob 0 -4. Kick up this is equal to minus minus four. I grew up less zero, Jacob less mm That is -4. Kick up. And let us suppose this vector is equal to be to cap. Then in order to find the normal vector to the plane You have to find the cross product of B one. And we do this is equal to Ichabod Jacob Jacob and The values are -3, -5 And -4, 0. And minus for therefore the after calculating this cross product, the normal vector will be equal to minus four Icap. Let's eat Jacob, my bliss for kick up. So from this we can calculate the equation on the plane as follows. That is Yeah. Yeah. By the formula To find the equation of the planet is in two X minus egg zero plus being two Y minus Y zero Plus seen to that zero equal to zero, bear A. B. C. Although mhm coefficient of direction vectors for the normal victor. That is this is A. This is B. This is C. And egg zero, Y. Zero and zero. Although point in the Mhm len here this point is equal to 314 as given in the question. Okay so the aggression of the plane is equal to yeah minus four X minus three. Let's eat. Why? Minus one last four. That minus four Is equal to zero. After solving the situation, we can derive the final equation of the plane as minus X plus two. Y. Let's said equal to three. And this is the required equation under lean for the given question.

Question on the plan to study Container Point Next zero was zero and a number of it and the co two a B C. And then we have the question will be on perform any times x minus X zero. Let's be times why minus y zero. Let's see Times Z minus zero because 20 in this question were given the plan that it contains Opponent 535 and it has number of it and ego to the two I, plus Shame and Escape. Notice that we can really stand in the back to perform Thio one minus one that for the equation of plan we will have it would be two times X minus five plus one times Y minus three once one time C minus five, equal to zero. And we soon. If I got a two x minus 10 plus y minus three minus C plus five equal to zero, I'll get it to X plus y minus C. They were equal to this will be honest 13 and then minus. It will be eager to eat here


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