Determine if a point is within two planes

Question

I have a point P and two planes defined by three vertex each. How can I determine if P is between the two planes?

Answer 1

"Between" only makes sense if the two planes are parallel, in wich case they are fully determined by the one plane and the (signed) distance $h$ of the other plane. Then compute the signed distance $d$ of your point to the first plane and compare these two values.
If $|d| < |h|$ and ${\rm sgn}(d) = {\rm sgn}(h)$, the point is between the two planes, else it is not.

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To find $h$ and $d$, chose the first plane and bring it into normal form, that is $$P_1 = \{x\in\mathbb R^3 | \langle x, \theta\rangle =s\}$$ Where $s \in\mathbb R$ and $\theta\in S^2$ is a unit vector (the orientation is here).
Now take a Point $p_2$ from $P_2$ (any of the three given, assuming you know that $P_1 \parallel P_2$) and you get $$P_2 = \{x\in\mathbb R^3 | \langle x, \theta\rangle = \langle p_2, \theta\rangle\}$$ We have found $h = \langle p_2, \theta\rangle - s$.
Similarly you get $d = \langle p, \theta\rangle - s$ where $p$ is you point.
$\langle\cdot,\cdot\rangle$ denotes the euclidean inner product.