Volume of parallelopiped
What is the volume of a parallelopiped with adjacent vertices at O(0, 0, 0), A(1, 1, -2), B(3, 1, -1) and C(2, -1, 1)?
I got the vectors:
OA = (1,1,-2)
OB = (3,1,-1)
OC = (2,-1,1)
but I am not sure which 2 vectors I must cross (and obviously the other one will get dotted). How do you know which 2 vectors are in the same plane in order to cross them?
It doesn't matter! and the other 5 permutations of the vectors differ only in sign. Since the sign does not matter.
In fact, with , , and , since , it is easy to see that .
Permuting the vectors just permutes the rows which only changes the sign. And, again, the volume is the absolute value of that.
(Three points determine a plane. Any two vectors, having the same "initial point", lie in a plane.)