Find symmetric equations for the line of intersection of the planes.
$ 5x - 2y - 2z = 1 $ , $ 4x + y + z = 6 $
$x=1, \quad y-2=-z$
in the question they're asking to find this metric equations for the lines of intersection of the plains. Given by the equation 15 x minus two, Y minus two is equal to one and two for express fibers equal to six. So in order to find those imagery equations for the line of intersection of these planes first we have to find the normal vectors of each of these planes. So the normal vector in one for the plane one is equal to 5 -2 -2. And the normal vector enter for the plane to is equal to 411. Therefore, in order to find the perpendicular to the line of intersection these planes, we have to find the cross product of these two normal vectors. That is the normal vector of the line of intersection of these two planes that is equal to given by the table for calculating the cross, productive The two normal vectors and one is 5 -2 -2 And into his 4 1. 1. This is equal to zero. Icap minus hurting Jacob blessed hurting Geico and therefore n vector is equal to 0 -13 and hurting. Now a point in this line of intersection of these planes can be found by putting any value to uh variable Let Z is equal to zero. Therefore in equation one it becomes five x minus two, Y is equal to one. Any question to becomes four X. Bless why Is equal to six. In order to calculate the value of X and Y. From the simultaneous equations. We Multiply the second equation by two and add these equations. Therefore the value will become after multiplying eight, X Plus two, Y is equal to 12. Therefore after adding these equations we get 13 x vitamins will get cancelled out Is equal to 13. Therefore from this we get X values equal to one and putting X equal to one. In any of these equations, we get five into one plus minus two. Y is equal to one. That is from equation one we get value of Y is equal to two. Thank you. Therefore we found the points the coordinates of the point of intersection of these planes. Yeah, mm hmm. That is equal to 1 to 0. And in order to find the symmetric equation of yeah, the line of intersection Of these two planes. Mm Okay. Yeah. The formula is X coordinate minus The coordinator at the point of intersection of these two plane. For the x coordinate, that is one. I'm divided by the normal victor or X coordinate had since This is equal to zero. Therefore we put X equal to one For the equation. And similarly or the white coordinated is why -2 divided by -13. And for that coordinated is said zero divided by 13. Since these two values can be put in the regular form, these two can be equated and therefore after simplifying this equation, we get the value as why minus two is equal to minus said. Therefore though symmetric equation or the line of intersection of these planes is okay, X equal to one, and Why -2 is equal to minus, it and this is the required to answer the given question.