Question

Find the vector and cartesian equations of the plane passing through the points $A(1,1,-2),B(1,2,1)$ and $C(2,-1,1)$

Answer 1

Given that plane passes through the points $A(1,1,-2),B(1,2,1),C(2,-1,1)$

The vectors $BA=j+3k$ and $CA=i-2j+3k$ lies on the given plane

So the vector $BA\times CA$ will be perpendicular to given plane

$\Rightarrow n=BA\times CA=9i+3j-k$

Equation of plane is

$r.n=a.n$

The plane through point $a=A(1,1,-2)$(say),$r=xi+yj+zk$

$a.n=14$

Therefore the equation of the plane in Cartesian form is $9x+3y-z=14$