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Thread: Volume of parallelopiped

  1. #1
    Apr 2010

    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?
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  2. #2
    MHF Contributor

    Apr 2005
    It doesn't matter! $\displaystyle \vec{u}\cdot(\vec{v}\times\vec{w})$ and the other 5 permutations of the vectors differ only in sign. Since $\displaystyle volume= \left|\vec{u}\cdot(\vec{v}\times\vec{w})\right|$ the sign does not matter.

    In fact, with $\displaystyle \vec{u}= x_1\vec{i}+ \y_1\vec{j}+ z_1\vec{3}$, $\displaystyle \vec{v}= x_2\vec{i}+ \y_2\vec{j}+ z_2\vec{k}$, and $\displaystyle \vec{w}= x_3\vec{i}+ y_3\vec{j}+ z_3\vec{3}$, since $\displaystyle \vec{v}\times\vec{w})= \left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3\end{array}\right|$, it is easy to see that $\displaystyle \vec{u}\cdot(\vec{v}\times\vec{w})= \left|\begin{array}{ccc}x_1 & y_1 & z_ 1\\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3\end{array}\right|$.

    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.)
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