Question

Find the angle between the plane passing through points $(1,1,1),(1,-1,1),(-7,-3,-5)$ and x-z plane

Answer 1

$A(1,1,1),B(1,-1,1),C(-7,-3,-5)$

$x-z$ plane $\Rightarrow y=0$

Equation of plane through $3$ point $\left(x_i{y}_i{z}_i\right),(i=1,2,3)$

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

Similarly,Equation of plane through $A,B,C$

$\left|\begin{array}{ccc}x-1& y-1& z-1\\ 0& 2& 0\\ 8& 2& 6\end{array}\right|=0\Rightarrow 3x-4z+1=0$

Angle between plane $3x-4z+1$ & $x-z$ plane be $\theta$.

$\therefore \cos \theta =0\Rightarrow \theta ={90}^{\circ }$