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Find the vector equation of the plane th...

Find the vector equation of the plane through the line of intersection of the planes `x+y+z=1\ a n d\ 2x+3y+4z=5` which is perpendicular to the plane `x-y+z=0.`

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Let equation of plane thorugh the the intersection of planes `x+y+z+=1` and `2x+3y+4z=5` is
`(x+y+z-1)+lambda(2x+3y+4z-5) = 0`
`rArr x(2lambda+1)+y(3lambda+1)+z(4lambda+1)-(1+5lambda)=0"……"`
It is perpendicular to the plane `x-y+z=0`
`:. 1(2lambda+1)-1(3lambda+1)+1(4lambda+1)=0`
`rArr 2lambda+1-3lambda-1+4lambda+1=0`
`rArr lambda = (-1)/(3)`
Put this value in equation (1)
`x((-2)/3+1)+y(-1+1)+z(-(4)/(3)+1)-(1-(5)/(3)) = 0`
`rArr x - z + 2 =0`
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