# Find the Vector Equation of the Plane Passing Through the Points i=j-2k, i+2j+k, 2i-j+k Hence Find the Cartesian Equation of the Plane. - Mathematics and Statistics

Find the vector equation of the plane passing through the points hati +hatj-2hatk, hati+2hatj+hatk,2hati-hatj+hatk. Hence find the cartesian equation of the plane.

#### Solution

Let

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

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

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

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

=9hati+3hatj-hatk

Then the equation of required plane is,

barr.barn=bara.barn

barr.(9hati+3hatj-hatk)=(hati+hatj-2hatk).(9hati+3hatj-hatk)

barr.(9hati+3hatj-hatk)=9+3+2

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

The cartesian equation of the plane is given by,

(xhati+yhatj+zhatk).(9hati+3hatj-hatk)=14,

9x+3y-z=14

The cartesian equation of the plane is 9x + 3y - z = 14.

Concept: Vector and Cartesian Equation of a Plane
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