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Find the equation the plane which con...

Find the equation the plane which contain the line of intersection of the planes` vec rdot( hat i+2 hat j+3 hat k)-4=0a n d vec rdot(2 hat i+ hat j- hat k)+5=0` and which is perpendicular to the plane ` vec r(5 hat i+3 hat j-6 hat k)+8=0` .

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The equation of the given planes are
`" "vecr*(hati+2hatj+3hatk)-4=0" "`(i)
`" "vecr*(2hati+hatj-hatk)+5=0" "` (ii)
The equation of the plane passing through the line of intersection of the plane given in (i) and (ii) is
`" "[vecr*(veci+2vecj+3veck)-4]+lamda[vecr*(2hati+hatj-hatk)+5]=0`
`" "vecr*[(2lamda+1)hati+(lamda+2)hatj+(3-lamda)hatk]+(5lamda-4)=0" "`(iii)
The plane in (iii) is perpendicular to the plane, `vecr*(5hati+3hatj-6hatk)+8=0`. Thus,
`" "5(2lamda+1)+3(lamda+2)-6(3-lamda)=0`
or `" "19 lamda-7=0`
or `" "lamda=(7)/(19)`
Substituting `lamda=(7)/(19)` in (iii), we get
`" "vecr*[(33)/(19)hati+(45)/(19)hatj+(50)/(19)hatk]-(41)/(19)=0`
or `" "vecr*(33hati+45hatj+50hatk)-41=0`
or `" "33x+45y+50z-41=0`
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