Question

Find the equation of the plane passing through the following point

 (1, 1, 1), (1, −1, 2) and (−2, −2, 2)

Answer 1

 The equation of the plane passing through points  (1, 1, 1), (1, −1, 2) and (−2, −2, 2) is given by

\[\begin{vmatrix}x - 1 & y - 1 & z - 1 \\ 1 - 1 & - 1 - 1 & 2 - 1 \\ - 2 - 1 & - 2 - 1 & 2 - 1\end{vmatrix} = 0\]

\[\]

\[ \Rightarrow \begin{vmatrix}x - 1 & y - 1 & z - 1 \\ 0 & - 2 & 1 \\ - 3 & - 3 & 1\end{vmatrix} = 0\]

\[\]

\[ \Rightarrow 1 \left( x - 1 \right) - 3 \left( y - 1 \right) - 6 \left( z - 1 \right) = 0\]

\[ \Rightarrow x - 1 - 3y + 3 - 6z + 6 = 0\]

\[ \Rightarrow x - 3y - 6z + 8 = 0\]