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.
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]|`
the vector equation of the given plane is scalar product form is
If ` barr =xhati+yhatj+zhatk, ` then the above equation becomes,
`∴ -y+2z=2 ` This is the cartesian form of the equation of required plane