What does $x_p y_q - x_q y_p$ represent in this vector function? When referring to someone else's formulae on doing vector calculations, I see them using the following function:
$$f(p, q) = x_p y_q - x_q y_p$$
...where $p$ and $q$ are two-dimensional points.
Unfortunately they don't say what they're actually trying to calculate here. I need to generalise this into three dimensions, so I need to understanding what they're doing; what is $f$?
Update:
Okay, apparently my question is not answerable without more context. Or rather, it is answerable, just not usefully. I appear to have tried too hard cutting out irrelevant detail.
The expression above is actually being used as follows:
$$g(p, q) = \sqrt{| |p-q|^2 - f(p, q)^2 |}$$
It's part of the equation of hyperbolic distance for the Beltrami-Klein model of hyperbolic space. $p$ and $q$ are both specified as 2D points in Euclidean space.
 A: $f(p,q)$ is the (signed) area of the parallelogram spanned by $p$ and $q$.  If you want to generalize this to three (and higher) dimensions, the key word is determinant.
A: I can't be sure because I don't know the context, but if $\overrightarrow {OP}=\left(\begin{matrix}x_p\\y_p\\0\end{matrix}\right)$ and $\overrightarrow {OQ}=\left(\begin{matrix}x_q\\y_q\\0\end{matrix}\right)$, then this is the $z$ component of the cross product of these two vectors, and is therefore the vector area formed by these.
A: Take the following two $3D$ vectors: $\overline p=(x_p,y_p,0)$ and $\overline q=(x_q,y_q,0)$. Then
$$\overline p \times \overline q=(0,0,x_py_q-y_px_q)$$
so $f(p,q)$ is the $z$ component of the vectorial product of the two vectors $\overline p$ and $\overline q$.
Update
$f(p,q)$ appears when one wants to determine the intersection points of the unit circle and the straight line determined by $p$ and $q$. Such a calculation is needed when one wants to calculate the hyperbolic length of the segment $pq$. (I assume that $p,q \in \text{ unit circle}.$)
