Given the eight vertices $(0,0,0)$, $(3,0,0)$, $(0,5,1)$, $(3,5,1)$, $(2,0,5)$, $(5,0,5)$, $(2,5,6)$, and $(5,5,6)$, find the volume of the parallelepiped.
I'm having trouble finding the 1 vertex and 3 vectors needed to find the volume. The closest four vertexes I found so far are $(0,0,0), (3,0,0), (0,5,1), (3,5,1)$…is using those four vertexes correct? Any starting hints to point me in the right direction?
Best Answer
Hint: If the origo is among them, the set of vertices of a parallelepiped is of the form $$\{0,\ a,\ b,\ c,\ a+b,\ a+c,\ b+c,\ a+b+c\}$$ for some vectors $a,b,c$.
Then write the coordinates of these $a,b,c$ in a matrix and calculate its determinant.