Question

Find equation of the plane passing through the points $P(1,1,1),Q(3,-1,2),R(-3,5,-4)$ ?

Answer 1

We know that,

Equation of plane passing through three non collinear,

Point...$(x_{1,}{y}_1,z_1)(x_2{y}_2,z_2)(x_{3,}{y}_3,z_3)$ is

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

Here,given points are $P(1,1,1)Q(3,-1,2)R(-3,5,-4)$

So equation of plane is

$\begin{array}{c}\left|\begin{array}{ccc}x-1& y-1& z-1\\ 3-1& -1-1& 2-1_1\\ -3-1& 5-1& -4-1\end{array}\right|=0\\ \\ \left|\begin{array}{ccc}x-1& y-1& z-1\\ 2& -2& 1\\ -4& 4& -5\end{array}\right|=0\end{array}$

$\begin{array}{c}(x-1)(10-4)-(y-1)(-10+4)+(z+1)(8-8)=0\\ (x-1)6-(y-1)(-6)+(z+1)0=0\\ (x-1)6-(y-1)(-6)=0\\ 6\left|(x-1)+(y-1)\right|=0\\ (x-1)+(y-1)=0\\ x+y=2\end{array}$