Question

Find the equations of the planes that passes through $(1,1,0),(1,2,1),(-2,2,-1)$

Answer 1

It is known that the equation of the plane through the points $(x_1,y_1,z_1),(x_2,y_2,z_2)$ and $(x_3,y_3,z_3)$ is,

$\left|\begin{array}{ccc}x-x_1& y-y_1& z-z_1\\ x_2-x_1& y_2-y_1& z_2-z_1\\ x_3-x_1& y_3-y_1& z_3-z_1\end{array}\right|=0$

$\Rightarrow$ $\left|\begin{array}{ccc}x-1& y-1& z\\ 0& 1& 1\\ -3& 1& -1\end{array}\right|=0$

$\Rightarrow$ $(-2)(x-1)-3(y-1)+3z=0$

$\Rightarrow$ $-2x-3y+3z+2+3=0$

$\Rightarrow$ $-2x-3y+3z=-5$

$\Rightarrow$ $2x+3y-3z=5$

This is the Cartesian equation of the required plane.