Find the vector equation of the plane passing through the intersection of the planes `x-2y+z=1a n d2x+y+z=8` and parallel to the line with direction ratios proportional to `1,2,1.` Find also the perpendicular distance of `(1,1,1)` from this plane.

Answer

`9x-8y+7z=21,13/sqrt(194)` units

Explanation

Let the required plane be
`(x-2y+z-1)+ lambda(2x+y+z-8)=0`
`ltimplies(1+2lambda)x+(lambda-2)y+(1+lambda)z-(1+8lambda)=0`……………(i)
The d.r.'s of the normal to this plane are
`(1+2lambda),(lambda-2),(1+lambda)`.
The normal to the plane (i) is perpendicular to the line with d.r.'s 1,2,1.
`therefore (1+2lambda)+2(lambda-2)+(1+lambda).`
The normal to the plane (i) is perpendicular to the line with d.r's 1,2,1.
`therefore (1+2lambda)+2(lambda-2)+(1+lambda)=0 rArr lambda=2/5`.