I have to solve the following exercise. Let $P_0, P_1, P_2$ three not aligned points. What figure does the set of points $P$ such that $$ (P-P_0)\times(P_1-P_0)\cdot(P_2-P_0)=h $$ represent? And what figure does the condition $$ |(P-P_0)\times(P_1-P_0)\cdot(P_2-P_0)|=|h| $$ represent?
My attempt. The height of the parallelepiped with sides $P_1-P_0$, $P_2-P_0$ and $P-P_0$ is fixed and it is equal to $$ (\star)\ \ \ \ \ \ \ \ \ \ \ height=\frac{|h|}{|(P_1-P_0)\times(P_2-P_0)|},\quad h=\text{volume of the parallelepiped}. $$ On the other hand, the height coincides with the distance between $P$ and $\pi$ = the plane containing $P_0, P_1$ and $P_2$. So I conclude that the required figure is a plane parallel to $\pi$. But what about the figure represented by all $P$ such that $$ |(P-P_0)\times(P_1-P_0)\cdot(P_2-P_0)|=|h|? $$ Some hints?
Thank You
The absolute value corresponds to $h$ or $(P-P_0)\times(P_1-P_0)\cdot(P_2-P_0)$ being either positive or negative, so it represents two planes, one $h$ units above $\pi$ and one $h$ units below.