Question

Find the equation of the plane passing through the points $(1,1,1),(3,-1,2),(-3,5,-4)$.

Answer 1

let $(x_1,y_1,z_1)=(1,1,1,)$

$(x_2,y_2,z_2)=(3,-1,2)$

and$(x_3,y_3,z_3)=(-3,5,-4)$

Equation of plane is given by

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

$\left|\begin{array}{ccc}x-1& y-1& 3-1\\ z-1& -1-1& 2-1\\ -3-1& 5-1& -4-1\end{array}\right|=0$

$\left|\begin{array}{ccc}x-1& y-1& z-1\\ 2& -2& 1\\ -4& 4& -5\end{array}\right|=0$

$(x-1)(10-4)+(y-1)(-4+10)+(z-1)(8-8)=0$

$6(x-1)+6(y-1)=0$

$x-1+y-1=0$

$x+y=2$