Check if a point is inside a rectangular shaped area (3D)? I am having a hard time figuring out if a 3D point lies in a cuboid (like the one in the picture below). I found a lot of examples to check if a point lies inside a rectangle in a 2D space for example this on but none for 3D space. 

I have a cuboid in 3D space. This cuboid can be of any size and can have any rotation. I can calculate the vertices $P_1$ to $P_8$ of the cuboid. 
Can anyone point me in a direction on how to determine if a point lies inside the cuboid? 
 A: The three important directions are $u=P_1-P_2$, $v=P_1-P_4$ and $w=P_1-P_5$.  They are three perpendicular edges of the rectangular box.
A point $x$ lies within the box when the three following constraints are respected:


*

*The dot product $u.x$ is between $u.P_1$ and $u.P_2$  

*The dot product $v.x$ is between $v.P_1$ and $v.P_4$  

*The dot product $w.x$ is between $w.P_1$ and $w.P_5$  



EDIT:
If the edges are not perpendicular, you need vectors that are perpendicular to the faces of the box.  Using the cross-product, you can obtain them easily:
$$u=(P_1-P_4)\times(P_1-P_5)\\
v=(P_1-P_2)\times(P_1-P_5)\\
w=(P_1-P_2)\times(P_1-P_4)$$
then check the dot-products as before.
A: Given $p_1,p_2,p_4,p_5$ vertices of your cuboid, and $p_v$ the point to test for intersection with the cuboid, compute:
$$\begin{matrix}
i=p_2-p_1\\ 
j=p_4-p_1\\ 
k=p_5-p_1\\
v=p_v-p_1\\
\end{matrix}$$
then, if
$$\begin{matrix}
0<v\cdot i<i\cdot i\\
0<v\cdot j<j\cdot j\\
0<v\cdot k<k\cdot k
\end{matrix}$$
The point is within the cuboid.
A: Since 2-D problem is known, divide the problem into 3 parts.
Disregard z-coordinate. If any point is within box bounds $ x_2-x_1,y_2-y_1, $ select it and all other such points. Similarly y-z and z-x boxes.
Next choose points that the are common to three selections.
