Find the vector and cartesiooan equations of the plane passing through the points `A(1,1-2),B(1,2,1),C (2,-1,1)`

Explanation

Let `bara, barb, barc`, the p.v. of the points `A(1,1,-2) B (1,2,1),C(2,-1,1)` respectively.
`:. Bara=hati+hatj-2hatk`
`barb=hatj+2hatj+hatk`
`bar=2hati+hatj+hatk`
`:. Bar(AB)= barb-barb=(2hati-hatj+hatk)-(hati+hatj-2hatk)`
`hatj+3hatk`
`bar(AC)=barc-bara=(2hati-hatj+hatk)-(ahti+hatj-2hatk)`
`= hati-2hatj+3hatk`
`:. bar(AB)xxbar(AC)=|{:(hati,hatj,hatk),(0,1,3),(1,-2,3):}|`
`= hati(3+6)-hatj(-3)+hatk(-1)`
`=9hati+3hatj-hatk`
The vector equation of the plane passing through the given points is
`bar r (bar (AB)xxbar(AC)=bara(bar(AB)xxbar(AC))`
`:. barr (9hati+3hatj-hatk)=(hati+hatj-2hatk)(9hati+3hatj-hatk)`
`:. vec r (9hati+3hatj-hatk)=9+3+2`
` bar r(9hati+3hatj-hatk)=14 " " ....(1)`
For cartesian form,
Putting `bar=xhati+yhatj+zhat` in (1), we get
`(x hati+yhati+zhatk)(9hati+3hatj-hatk)=14`
`:. 9x+3y-z=14`
`:.` Required cartesian form of the euquation of the plane is `9x+3y-z=14`