Find an equation of the plane.
The plane that passes through the point $ (3, 1, 4) $ and contains the line of intersection of the planes $ x + 2y + 3z = 1 $ and $ 2x - y + z = -3 $
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.