The base of a pyramid is a rectangle with corners in $(1, 0, 2), (2, 1, 0), (0, 1, 2)$ and $(0,0,3)$. The top has the coordinates $(7, 3, 5)$. Calculate the volume of the pyramid. The coordinates are set in an orthonormal basis.
I am thinking that I first calculate three vectors that make up three sides of the pyramid, from one starting point, like so:
$u= (7, 3, 5)-(0, 0, 3) = (7, 3, 2)$
$v = (0, 1, 2) - (0, 0, 3) = (0, 1, -1)$
$w = (2, 1, 0) - (0, 0, 3) = (2, 1, -3)$
after that I calculate $\det A$, where $u,v$ and $w$ are column matrices. $\det A = 24$. Because it is a pyramid and not a parallelepiped, I need to divide it. The formula for a pyramid is $\frac{\text{base}\cdot\text{height}}{3}$, right? So the volume according to my calculations would be $8vu$, but that is inaccurate, because the right answer is $6vu$. Therefore, I could adjust my answer to divide $\det A$ by $4$, but I don't understand why I would do that and I suspect it would be a faulty procedure. Would anybody care to explain what it is I'm doing wrong?