I have $P=(p_1,p_2)$ and $Q=(q_1,q_2$) two points in $\mathbb R^2$, $P\ne Q$, and $R=(r_1,r_2)$ another point. What means the following determinant?
$$\Delta (P, Q, R)=
\begin{vmatrix}
1 & 1 & 1 \\
p_1 & q_1 & r_1 \\
p_2 & q_2 & r_2 \\
\end{vmatrix}
$$
Best Answer
That is twice the signed area of the triangle $\triangle PQR$. See this for some more details.
By "signed" I mean that the vertices must be taken in the counterclockwise direction to get the area. If taken clockwise, you get the negative of the area.