Show that the following points are coplanar:
$(0, - 1, 0), (2, 1, - 1), (1, 1, 1)$ and $(3, 3, 0)$

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Solution

Let the given points are
$A (0, - 1, 0), B (2, 1, - 1), C (1, 1, 1), D (3, 3, 0)$
$- - \to A B = - - \to O B - - - \to O A = 2^i + 2^j -^k$
$- - \to A C = - - \to O C - - - \to O A =^i + 2^j +^k$
$- - \to A D = - - \to O D - - - \to O A = 3^i + 4^j + 0^k$
$[\begin{matrix} - - \to A B & - - \to A C & - - \to A D \end{matrix}] = ∣ ∣ ∣ ∣ \begin{matrix} 2 & 2 & - 1 1 & 2 & 1 3 & 4 & 0 \end{matrix} ∣ ∣ ∣ ∣$
$= 2 (0 - 4) - 2 (0 - 3) - 1 (4 - 6) = 0$
Therefore points $A, B, C$ and $D$ are coplanar.

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