Let points $(0,0,0), (1,2,x), (-2,1,0)$ and $(1,1,3)$ be at four corners of a parallelpiped. Determine
the volume of the parallelepiped by using the determinant in terms of $x$. For what value of $x$ is
the volume $0$ . Interpret this case geometrically.
I'm new in linear algebra.
how just 4 points can define a parallelepiped ?
and how i can value of $x$ that will make volume $0$ ?
any hint?
Best Answer
Maybe a picture helps (from Wikipedia):
If $a=(1,2,x)$, $b=(-2,1,0)$, $c=(1,1,3)$, then this is your parallelpiped.
You can convince yourself, that its volume is given by the determinant:
$$\left|\det\left(\begin{matrix}1 & -2 & 1\\2 & 1 & 1\\x & 0 & 3\end{matrix}\right)\right|$$