Problem 67 Hard Difficulty

Which of the following four planes are parallel? Are any of them identical?

$ P_1 : 3x + 6y - 3z = 6 $ $ P_2 : 4x - 12y + 8z = 5 $
$ P_3 : 9y = 1 + 3x + 6z $ $ P_4 : z = x + 2y - 2 $

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WZ

Wen Z.

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Video Transcript

in the question they're asking that each of the planes P one P two, P three before our marlin. Yeah. And if they are panel then which of them are identical. Yeah. So which Everyone is given by the equation three explicit XY -3 is equal to six. P two is 4 X- to invite us. It is equal to five. History is nine by equal to one plus three plus six and before is equal to zero is equal to express to one minus two. So in order to find the planes are parallel or not we have to find the ratio between the coordinate coordinates coefficients of the plane. So first We compare the values of P one and P two checking if they're panel or not. So the coefficients of the coordinates are P one is given by the equation So P one and P two is given by the equations. And so the coefficients of the coordinates are yeah computed in the form of ratio in order to find whether they are paddle or not. So three by four X coordinate is equal to six by minus two. Report by coordinate is equal to minus three by eight. Or that coordinate if we simplify this then it will be equal to three by four is not equal to minus half Is not equal to -3 x eight. Therefore Clearly P one and P two are not identical but are not parallel. Hence not identical. So this is the first case. Next we have to check for the value for the planes P one and beach three which is given by the equations. So the equations of P one and P three planes are this. And so for checking whether these are parallel or not, we have to compare the coefficients of the coordinates which is equal to three by 34. X coordinate is equal to six by minus nine. For the wife coordinate is equal to minus three by six. For the that coordinate that is equal to after simplifying we get the value as one is not equal to -2 x three is not equal to minus half. Therefore P one and P three R not parallel. Hence these are also not identical. Now in case three we have to check for the plains even and before. So the equations of planes, people and before are these therefore have to compare the coefficients of the coordinates that is equal to three by one or x coordinate equal to six by two. For y coordinate is equal to minus three by minus 1% coordinates. So after simplifying these ratios we get the value as equal to three equal to three equal to three. Therefore B one. And before our Berlin planes. And uh checking further whether whether they are identical or not. So we put the value of X equal to zero And by equal to zero. To find the value of said. So the equation for P one plane is three explicit Xy minus three is equal to six. Therefore The value of Z. Here is equal to -2. Therefore the point for this plane That is a point containing this plane is 00 -2. Similarly for the plane before the equation is express to why miners said is equal to um two. Therefore If we put the value of x equal to zero and by equal to zero then we get the value of that is -2. Therefore the point passing should this plane Is equal to 00 -2. Therefore B one and before are bought parlance and identical. Next case we consider for The plane Speed two and Petri. So the equations of planes P two and P three are these. So in order to check whether their parents are not here to compare the coefficients of the coordinates that is equal to four by 34 X coordinate equal to minus 12 and minus 94 by coordinate is equal to eight. Basics for the z coordinate. This is equal doctor simplifying these ratios. We get yeah the sequel to four bytes required four bytes required to four bytes. Three. Therefore see two and P three R. Berlin. And in order to check whether these planes are yeah identical or not let X equal to buy equal to zero. Therefore for the P two plane that will be equal to yeah five by eight. And uh So the point to this plane will be 005 x eight and same condition for the or plain Petri. So here there will be equal to -1 x six. So the point through this plane will be 00 -1 Way six. This is the value of five x 8 is equal to point six two. And the value of the that putting it off Point in Petri is -1. Basics. That is equal do minus point. Yeah. 16. Therefore the Point in P two is not equal to point in P three. Therefore P two and P three are not identical. The next case is you have to compare P two and P four. Yeah. So the equations of the plain speech win before are these? And therefore comparing these coefficients for the coordinates of these two planes to check whether their palates or not is equal to fall by one. For the x coordinate equal to minus 12 way too. For the y coordinate equal to eight by minus one for the z coordinate After simplifying This becomes equal to four is not equal to -6 is not equal to -8. Therefore B two and before I not yeah, Palin hence they are not identical also. Yeah. And uh last possibility is to check For P three and P four. So the equations of Patreon before our so the equations of P three and P four planes are these? Therefore we have to take whether these planes are talent or not. So comparing the ratios of the coordinates of these two planes is equal to three by one for X coordinate equal to minus nine by 24 by coordinated quarters six by -1 part. Is that fortunate? So these three are not equal hands P three and before are not talent, and hence not identical. Also, therefore, the planes that are pal an identical art. Therefore P two P 3 planes are Palin and not identical and before and people are both paddles and identical to each other. So this is the required answer of the given question.