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**Mush** Let your four points be A, B, C, D. To prove they are coplanar, find equations for lines AB, AC and AD. They will be of the form:

$\displaystyle AB = a\vec{i} + b \vec{j} + c\vec{k}$

$\displaystyle AC = d\vec{i} + e \vec{j} + f\vec{k}$

$\displaystyle AD = g\vec{i} + h \vec{j} + i\vec{k}$

Which can be written in matrix form:

$\displaystyle \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \begin{pmatrix} \vec{i} \\ \vec{j} \\ \vec{k} \end{pmatrix} $ (Not sure if that's much use though.

Points are coplanar if the triple product is 0.

Triple product involves the cross product of two of the above:

$\displaystyle AB \times AC = det \begin{pmatrix} \vec{i} & \vec{j} & \vec{k} \\ d & e & f \\ g & h & i \end{pmatrix} $