# 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). - Mathematics and Statistics

Short Note

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

Concept: Plane - Equation of Plane Passing Through the Given Point and Perpendicular to Given Vector
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