How to check if point is within a rectangle on a plane in 3d space Please refer to this image for this question->

I have a 3d bounded box (in green).
I also have a 3d line (in red)
I know the points a, b, c, d, e. They are points in space with x, y, z, coordinates.
I also know that a, b, c, d, and e, lie on the same plane.
i.e. point a is the intersection between the plane of (b, c, d , e) with the red line.
The box is not axis aligned.
Now, what i want to know is, how can i calculate whether the point a, lies inside of (b, c, d, e) box? Obviously it doesn't in this case, but how can i calculate this?
They are on the same plane, so it should be a 2d problem, but all my coordinates are in 3d so i'm not sure how to do it. Can someone help?
This is not homework, i am doing some hobby game programming.
 A: If $b,c,d,e$ are a rectangle and $a$ is coplanar with them, you need only check that $\langle b, c-b\rangle\le \langle a, c-b\rangle\le \langle c, c-b\rangle$  and $\langle b, e-b\rangle\le \langle a, e-b\rangle\le \langle e, e-b\rangle$ (where $\langle,\rangle$ denotes scalar product).
A: Hint: one way to transform your 3D coordinates to 2D ones, for the purpose of this question.
Set any point on the given plane to be the origin, e.g. I choose B.
$$\begin{align}
\mathbb{a'} =& \mathbb{a}-\mathbb{b}\\
\mathbb{b'} =& \mathbb{b}-\mathbb{b} = 0\\
\mathbb{c'} =& \mathbb{c}-\mathbb{b}\\
\mathbb{d'} =& \mathbb{d}-\mathbb{b}\\
\mathbb{e'} =& \mathbb{e}-\mathbb{b}\\
\end{align}$$
Now, since BCDE is a rectangle, $\mathbb{c'}$ and $\mathbb{e'}$ are orthogonal to each other. View $\{\mathbb{c'},\mathbb{e'}\}$ as the basis of the plane.
And since A is on the same plane as BCDE, you can write $\mathbb{a'} = p\mathbb{c'} + q\mathbb{e'}$ form. Now, iff $p$ and $q$ are both within 0 and 1, point A is in BCDE.
