Condition for 4 Points to be Coplanar

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Theorem

Let:

\(\ds p_1\) \(=\) \(\ds \tuple {x_1, y_1, z_1}\)
\(\ds p_2\) \(=\) \(\ds \tuple {x_2, y_2, z_2}\)
\(\ds p_3\) \(=\) \(\ds \tuple {x_3, y_3, z_3}\)
\(\ds p_4\) \(=\) \(\ds \tuple {x_4, y_4, z_4}\)

be distinct points in Cartesian $3$-space.


Then $p_1$, $p_2$, $p_3$ and $p_4$ are coplanar if and only if:

$\begin {vmatrix} x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \\ x_4 & y_4 & z_4 & 1 \end {vmatrix} = 0$

where the construct is evaluated as a determinant.


Proof


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Sources

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