A plane can be taken to be a quadruple $(a, b, c, d)$. Then a point is taken to be a triple of planes $(P, Q, R)$. The point $(p, q, r)$ is valid if and only if the determinant of the matrix $A$ below is non-zero.
$$A= \left[ \begin{array}{ccc} Pa & Pb & Pc\\ Qa & Qb & Qc\\ Ra & Rb & Rc\\ \end{array} \right] $$
We can get it by simple linear algebra: The point $(P, Q, R)$ can be presented as the equation root of $Ax = b$. If this is valid then $Ax = b$ has an unique solution which implies $\det(A) \neq 0$.
Given that the point $(P, Q, R)$ is valid, it lies behind, on, or in-front of the plane $S$ if and only if the following expression is negative, zero or positve.
$$ \left\vert \begin{array}{cccc} Pa & Pb & Pc & Pd\\ Qa & Qb & Qc & Qd\\ Ra & Rb & Rc & Rd\\ Sa & Sb & Sc & Sd \end{array} \right\vert \left\vert \begin{array}{ccc} Pa & Pb & Pc\\ Qa & Qb & Qc\\ Ra & Rb & Rc \end{array} \right\vert $$
My question is: What is the mechanism behind this?