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

#### Solution

The vector equation of the plane passing through the points `A(bara),B(barb)andC(barc)`

`barr.(bar(AB)xxbar(AC))=bara(bar(AB)xxbar(AC))`............(1)

Let `bara=hati+hatj-2hatk,barb=hati+2hatj+hatk,barc=2hati-hatj+hatk`

`therefore bar(AB)=barb-bara=(hati+2hatj+hatk)-(hati+hatj-2hatk)=hatj+3hatk`

and `bar(AC)=barc-bara=(2hati-hatj+hatk)-(hati - hatj-2hatk)=hati-2hatj+3hatk`

`therefore bar(AB)xxbar(AC)=|[hati,hatj,hatk],[0,1,3],[1,-2,3]|`

`=(3+6)hati-(0-3)hatj+(0-1)hatk`

`=9hati+3hatj-hatk`

`bara.(bar(AB)xxbar(AC))=(hati+hatj-2hatk).(9hati+3hatj-hatk)`

`=1(9)+1(3)+(-2)(-1)`

`=9+3+2=14`

from (1), the vector equation of the required plane is

`barr.(9hati+3hatj-hatk)=14`

`(xhati+y hatj + z hatk)(9hati + 3hatj- hatk) = 14`

∴ the cartesian equation of the plane is

`9x + 3y -z = 14`