Find the vector of the plane passing through the intersection of the planes `vecr.(2hati+2hatj-3hatk)=7, vecr.(2hati+5hatj+3hatk)=9` and the point (2,1,3)`.

Explanation

The equations of the given planes are `vecr.(2hati+2hatj-3hatk)=7` and `vecr.(2hati+5hatj+3hatk)=9`.
These are of the form `vecr.(vecn_(1)+lambdavecn_(2))=(q_(1)+lambdaq_(2))` for some real number `lambda`.
`rArr vecr.[(2hati+2hatj-3hatk)+lambda(2hati+5hatj+3hatk)](7+9lambda)`
`rArr vecr.[(2+2lambda)hati(2+5lambda)hatj+(-3+3lambda)hatk]=(7+9lambda)`. ..................(i)
If the plane (i) passes through the point A(2,1,3), then
`vecr.(2hati+hatj+3hatk)` must satisfy it.
`therefore (2hati+hatj+3hatk).[2(2+2lambda)hati+(2+5lambda)hatj+(-3+3lambda)hatk]=(7+9lambda)`
`rArr vecr.[(2+2lambda)hati(2+5lambda)hatj+(-3+3lambda)hatk]=(7+9lambda)`........(i)
If the plane (i) passes through the point A(2,1,3), then
`vecr=(2hati+hatj+3hatk)` must satisfy it.
`therefore (2hati+hatj+3hatk).[(2+2lambda)hati+(2+5lambda)hatj+(-3+3lambda)hatk]=(7+9lambda)`.
`rArr 2(2+2lambda)+1.(2+5lambda)+3(-3+3lambda)=7+9lambda`.
`rArr (4+4lambda)+(2+5lambda)+(-9+9lambda)=7+9lambda rArr 9lambda=10 rArr lambda=10/9`.
Putting `lambda=10/9` in (i), we get the required equation of the plane as
`vecr.[(2+20/9)hati+(2+50/9)hatj+(-3+30/9)hatk]=(7+10)`.
`rArr vecr.(38hati+68hatj+3hatk)=153`.
Hence, the required equation of the plane is
`vecr.(38hati+68hatj+3hatk)=153`.