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Math Help - Parallelepiped Vertices

  1. #1
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    Parallelepiped Vertices

    If vector OA, vector OB, and vector OC are three edges of a parallelpiped where O is (0,0,0), A is (2,4,-2), B is (3,6,1) and C is (4,0,-1) find the coordinates of the other vertices of the parallelpiped.

    for this q, i am having trouble drawing the diagram of the parallelpiped

    Currently though, I have the following possibilities.. (there are only 3 right?)

    D(5,10,0)<<< DB//CA
    D(3,-2,0)<<<CB//DA
    D(1,10,0)<<<CB//AD
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  2. #2
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    Hello skeske1234
    Quote Originally Posted by skeske1234 View Post
    If vector OA, vector OB, and vector OC are three edges of a parallelpiped where O is (0,0,0), A is (2,4,-2), B is (3,6,1) and C is (4,0,-1) find the coordinates of the other vertices of the parallelpiped.

    for this q, i am having trouble drawing the diagram of the parallelpiped

    Currently though, I have the following possibilities.. (there are only 3 right?)

    D(5,10,0)<<< DB//CA
    D(3,-2,0)<<<CB//DA
    D(1,10,0)<<<CB//AD
    If you're having trouble drawing it, then don't try! Just think of it as being a boring old cuboid (rectangular box) that's been squashed a bit.

    Instead of rectangles, then, the faces are all parallelograms. So if we consider the face containing O, A and B, the fourth vertex of this parallelogram - P, say - is such that:

    \vec{OP} = \vec{OA} + \vec{OB} = \begin{pmatrix} 2\\4\\-2\end{pmatrix}+ \begin{pmatrix} 3\\6\\1\end{pmatrix}=\begin{pmatrix} 5\\10\\-1\end{pmatrix}

    Then deal with the 2 faces containing O, A and C; and O, B and C, respectively, in the same way to get, say, vertices Q and R.

    Finally, we shall need the eighth vertex, S, say, whose position vector is the sum of all three vectors \vec{OA} + \vec{OB}+\vec{OC}. This vertex will be diagonally opposite O.

    Can you complete it now?

    Grandad
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