Find the equation of the plane passing through `(a ,b ,c)` and parallel to the plane ` vec rdot( (hat i+ hat j+ hat k))=2.`

Explanation

As we know that any plane which is parallel to the plane `r .(hati+hatj​+hatk)=2` is for the form
`r .(hati+hatj+hatk)=λ.....(1)`
As this plane passes through the point `(a,b,c)`.
Therefore, the position vector r. of this point is `ahati+bhatj​+chatk Therefore, equation (1) becomes
`(ahati+bhatj+chatk)*(hati+hatj​+hatk)=λ`
`=>a+b+c=λ`
Substituting `λ=a+b+c` in equation (1) we obtain
`r *(hati+hatj+hatk)=a+b+c......(2)`
This is the vector equation of the required plane.
Substituting `r =x*hati+yhatj​+zhatk` in equation (2), we obtain
`(xhati+yhatj​+zhatk).(hati+hatj​+hatk)=a+b+c`
`=>x+y+z=a+b+c`