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

Here, apply the equation of the plane passing through the points $\left( {{x_1},{y_1},{z_1}} \right),\left( {{x_2},{y_2},{z_2}} \right)$ and $\left( {{x_3},{y_3},{z_3}} \right)$ is given by

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

As we know, 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 $\left( {{x_3},{y_3},{z_3}} \right)$ is

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

$\Rightarrow$ $\left| {\begin{array}{cccccccccccccccccccc}{x - 2}&{y - 1}&{z - 0}\\{3 - 2}&{ - 2 - 1}&{ - 2 - 0}\\{3 - 2}&{1 - 1}&{7 - 0}\end{array}} \right| = 0$

$\Rightarrow$ $\left| {\begin{array}{cccccccccccccccccccc}{x - 2}&{y - 1}&z\\1&{ - 3}&{ - 2}\\1&0&7\end{array}} \right| = 0$

$\Rightarrow$ $(x - 2)( - 21 + 0) - (y - 1)(7 + 2) + z(3) = 0$

$\Rightarrow$ $- 21x + 42 - 9y + 9 + 3z = 0$

$\Rightarrow$ $- 21x - 9y + 3z = - 51$
$\therefore$ $7x + 3y - z = 17$

So, the required equation of plane is $7x + 3y - z = 17$.