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Question
Show that the following points are coplanar.
(0, 4, 3), (−1, −5, −3), (−2, −2, 1) and (1, 1, −1)
Solution
(ii) The equation of the plane passing through (0, 4, 3), (−1, −5, −3), (−2, −2, 1) is
\[\begin{vmatrix}x - 0 & y - 4 & z - 3 \\ - 1 - 0 & - 5 - 4 & - 3 - 3 \\ - 2 - 0 & - 2 - 4 & 1 - 3\end{vmatrix} = 0\]
\[ \Rightarrow \begin{vmatrix}x & y - 4 & z - 3 \\ - 1 & - 9 & - 6 \\ - 2 & - 6 & - 2\end{vmatrix} = 0\]
\[ \Rightarrow - 18x + 10 \left( y - 4 \right) - 12 \left( z - 3 \right) = 0\]
\[ \Rightarrow 9x - 5 \left( y - 4 \right) + 6 \left( z - 3 \right) = 0\]
\[ \Rightarrow 9x - 5y + 6z + 2 = 0\]
\[\text{ Substituting the last point (1, 1, -1) (it means x = 1; y = 1; z=-1) in this plane equation, we get } \]
\[9 \left( 1 \right) - 5 \left( 1 \right) + 6 \left( - 1 \right) + 2 = 0\]
\[ \Rightarrow 4 - 4 = 0\]
\[ \Rightarrow 0 = 0\]
\[\text{ So, the plane equation is satisfied by the point (1, 1, -1) } .\]
\[\text{ So, the given points are coplanar } .\]
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