Find the vector equation of the plane passing through the point `hati + hatj -2hatk` and `hati + 2hatj + hatk, 2hati - hatj + hatk.`

Explanation

`veca, vecb, vecc` are the position vectors
i.e., `veca=hati+hatj-2hatk, vecb=hati+2hatj+hatk, vecc=2hati-hatj+hatkand vecb-veca=0hati+hatj+3hatk`
`vecc-veca=hati-2hatj+3hatk`
Now, `(vecb-veca)xx(vecc-veca)=|{:(hati, hatj, hatk),(0,1,3),(1,-2,3):}|`
`=hati(3+6)-hatj(0-3)+hatk(0-1)=9hati+3hatj-hatk`
Let `vecn =(vecb-veca)xx(vecc-veca)=9hati+3hatj-hatk`
Then, the vector equation of the required plane is
`vecr*vecn=veca*vecn`
`vecr(9hati+3hatj-hatk)=(hati+hatj-2hatk)(9hati+3hatj-hatk)`
`vecr(9hati+3hatj-hatk)=9+3+2`
`vecr(9hati+3hatj-hatk)=14`
The cartensian equation of the plane is given by
`(x hati+yhatj+zhatk)(9hati+3hatj-hatk)=14`
`because"["vecr=x hati+yhatj+z hatj"]"`
`9x+3y-x=14`