The equation of the plane through the points \((2,1,0),(3,2,-2)\) and \((3,1,7)\) is

We know that the equation of a plane passing
through three non-collinear points.
\(\left(x_1, y_1, z_1\right),\left(x_2, y_2, z_2\right)\) and \(\left(x_3, y_3, z_3\right)\)
\(\because \quad\left|\begin{array}{lll}x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1\end{array}\right|=0\)
\(\begin{aligned} & \Rightarrow \quad\left|\begin{array}{ccc}x-2 & y-1 & z-0 \\ 3-2 & 2-1 & -2-0 \\ 3-2 & 1-1 & 7-0\end{array}\right|=0 \\ & \Rightarrow \quad\left|\begin{array}{ccc}x-2 & y-1 & z \\ 1 & 1 & -2 \\ 1 & 0 & 7\end{array}\right|=0 \\ & \Rightarrow \quad(x-2)(7-0)-(y-1)(7+2)+z(0-1)=0 \\ & \Rightarrow \quad 7 x-14-(y-1) 9+(-z)=0 \\ & \Rightarrow \quad 7 x-9 y-z-5=0\end{aligned}\)