# Parametric Form of the Equation of the Plane r=(2i+k)+λi+μ(i+2j+k) λ And μ Are Parameters. Find Normal to the Plane and Hence Equation of the Plane in Normal Form. Write Its Cartesian Form. - Mathematics and Statistics

Parametric form of the equation of the plane is bar r=(2hati+hatk)+lambdahati+mu(hat i+2hatj+hatk) λ and μ are parameters. Find normal to the plane and hence equation of the plane in normal form. Write its Cartesian form.

#### Solution

The vector equation of the plane barr =bara+lambda bar b+mu barc in scalar product form is

barr.(bar b xxbarc)=bara.(barbxxbarc)

Here , bar a=2hati+hatk, barb=hati, barc=hati+2hatj+hatk

therefore barb xxbarc=|[hati,hatj,hatk],[1,0,0],[1,2,1]|

=(0-0)hati-(1-0)hatj+(2-0)hatk

=-hatj+2hatk

and bara.(barbxxbarc)=(2hati+hatk).(-hatj+2hatk)

=2(0)+0(-1)+1(2)=2

the vector equation of the given plane is scalar product form is

bar r.(-hatj+2hatk)=2

If  barr =xhati+yhatj+zhatk,  then the above equation becomes,

(xhati+yhatj+zhatk)(-hatj+2hatk)=2

x(0)+y(-1)+z(2)=2

∴ -y+2z=2  This is the cartesian form of the equation of required plane

Concept: Vector and Cartesian Equation of a Plane
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