"Find the volume of the parallelepiped by four vertices: $(0,1,0), (2,2,2), (0,3,0),$ and $(3,1,2)$.
I know the formula to find this volume is: $|\vec{a} \circ(\vec{b}\times \vec{c})|$, and I know how to carry out the computation to get the actual value. What I need to know is the process of how I set up the values of the vectors $\vec{a},\vec{b},$ and $\vec{c}$ using the given points?
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$\begingroup$The volume of a parallelepiped determined by the vectors a, b ,c (where a, b and c share the same initial point) is the magnitude of their scalar triple product:
Take four points as $P=(0,1,0),Q=(2,2,2),R=(0,3,0),S=(3,1,2)$ and find $$PQ=a=\langle2-0,2-1,2-0\rangle=\langle2,1,2\rangle$$$$PR=b=\langle0-0,3-1,0-0\rangle=\langle0,2,0\rangle$$$$PS=c=\langle3-0,1-1,2-0\rangle=\langle3,0,2\rangle$$Then find $|a\cdot(b\times c)|$
$\endgroup$ 2 $\begingroup$Translate the parallelepiped such that one of the vertices is the origin. Then the volume has not changed and you can use your formula.
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