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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).

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


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]|`






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


`(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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