Question

Show that the following points are coplanar.
 (0, −1, 0), (2, 1, −1), (1, 1, 1) and (3, 3, 0) 

Answer 1

 The equation of the plane passing through points (0, −1, 0), (2, 1, −1), (1, 1, 1) is given by 

\[\begin{vmatrix}x - 0 & y + 1 & z - 0 \\ 2 - 0 & 1 + 1 & - 1 - 0 \\ 1 - 0 & 1 + 1 & 1 - 0\end{vmatrix} = 0\]
\[ \Rightarrow \begin{vmatrix}x & y + 1 & z \\ 2 & 2 & - 1 \\ 1 & 2 & 1\end{vmatrix} = 0\]
\[ \Rightarrow 4x - 3 \left( y + 1 \right) + \text{ 2  z }= 0\]
\[ \Rightarrow 4x - 3y + 2z - 3 = 0\]
\[\text{ Substituting the last point (3, 3, 0) (it means x = 3; y = 3; z = 0) in this plane equation, we get } \]
\[4 \left( 3 \right) - 3 \left( 3 \right) + 2 \left( 0 \right) - 3 = 0\]
\[ \Rightarrow 12 - 12 = 0\]
\[ \Rightarrow 0 = 0\]
\[\text{ So, the plane equation is satisfied by the point (3, 3, 0). } \]
\[\text{ So, the given points are coplanar }.\]