This is an applied problem, which I try to define mathematically. I have two vehicles, vehicle 1 is defined by the area, dependent on length $L$ and width $W$ of the vehicle, according to: $\begin{bmatrix} -\frac{L}{2} \\[0.3em] -\frac{W}{2} \end{bmatrix} \leq A_{E} \leq \begin{bmatrix} \frac{L}{2} \\[0.3em] \frac{W}{2} \end{bmatrix} $
Globally, the middle point of this vehicle is positioned in point $E$, and the middle point on vehicle 2 is positioned in the global cartesian point $P$.
If we think about the problem as if everything is seen from vehicle 1's view, the length of vehicle 1 is aligned with the x-axis in the coordinate system. Vehicle 2 has an area made up of: $\begin{bmatrix} \mu_Px-\frac{\bar{L}}{2} \\[0.3em] \mu_Py-\frac{\bar{W}}{2} \end{bmatrix} \leq A_P \leq \begin{bmatrix} \mu_Px+\frac{\bar{L}}{2} \\[0.3em] \mu_Py+\frac{\bar{W}}{2} \end{bmatrix} $
where $\mu_Px$ and $\mu_Py$ are the mean of the x- and y coordinate of the measured position P.
In theory an infinite number of points are spread out over vehicle 2's area, but in reality we use N points. These points are rotated according to both vehicles headings. In the deterministic case we want to check whether any of the points of vehicle 2 is inside the (global) area of vehicle 1. This means that we need to find the difference between each point and evaluate if it is located in the bounding box $A_E$. For simplicity, consider N to be 1 and that this point is rotated according to the headings and denoted $P1$. The distance (vector?) between these points are calculated as:
$\overrightarrow{EP1} = [P1_x-E_x,P1_y-E_y]$
What I now want to check is if there is a risk of collision between the two vehicles (these two points, the problem is simplified) by evaluating the following:
$ P_{collision} = \begin{cases} 1 & \quad \text{if } \; \overrightarrow{EP1} \in A_{E}\\ 0 & \quad \text{otherwise } \end{cases} $
for example a collision risk will be present if E = (L/2,W/2) and P1 = (L/4,W/4)=>$\overrightarrow{EP1}=(L/4,w/4)$ which is inside $A_E$.
Now my question is how you should write and refere to this correctly, if it is correct to say that a vector ($\overrightarrow{EP1}$) belongs to an area, or if this should be referred to in some other way (maybe we should ask if a point belongs to the area?).
Here is some additional information than maybe can be of help, or only be confusing...
The measured and predicted positions of both vehicles are the global cartesian middle point of each vehicle and can be seen as two bivariate variables, Vehicle 1: $E\sim\mathcal{N}(\boldsymbol{\mu_E},\boldsymbol{\Sigma_E})$
and vehicle 2: $P\sim\mathcal{N}(\boldsymbol{\mu_P},\boldsymbol{\Sigma_P})$
The rotated point on vehicle 2 is also considered a bivariate variable denoted: $P_1\sim\mathcal{N}(\boldsymbol{\mu_{P_1}},\boldsymbol{\Sigma_P})$
By using linear combination of the variables similarly to the deterministic equation the following distribution can be formulated: $D\sim\mathcal{N}(\boldsymbol{\mu_{P_1}}-\boldsymbol{\mu_E},\boldsymbol{\Sigma_{E}}+\boldsymbol{\Sigma_P})$
with the use of the joint cumulative distribution can the probability of collision be calculated as, $ P=\iint\limits_{A_E} D\, \mathrm{d}x \, \mathrm{d}y = \int\limits_{-L/2}^{L/2}\int\limits_{-w/2}^{w/2} D\, \mathrm{d}x \, \mathrm{d}y $