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Math Help - Hard vector parallelepped question

  1. #1
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    Hard vector parallelepped question

    Three points A, B and C have coordinates (2,1,-2), (2,-1,-1) and (1,2,2) respectively. The vectors OA, OB and OC where O is theorigin form three concurrent edges of a parallelepiped OAPBCQSR in the diagram:

    http://i45.tinypic.com/1ordkp.jpg

    1. Find the coordinates of P, Q, R and S
    2. Fina an equation for the plan OAPB
    3. calculate the volume, B or the parallelepiped

    Any help appreciated, im not sure where to start for Q1 and Q2

    Thanks
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  2. #2
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    Hello Jampop
    Quote Originally Posted by Jampop View Post
    Three points A, B and C have coordinates (2,1,-2), (2,-1,-1) and (1,2,2) respectively. The vectors OA, OB and OC where O is theorigin form three concurrent edges of a parallelepiped OAPBCQSR in the diagram:

    http://i45.tinypic.com/1ordkp.jpg

    1. Find the coordinates of P, Q, R and S
    2. Fina an equation for the plan OAPB
    3. calculate the volume, B or the parallelepiped

    Any help appreciated, im not sure where to start for Q1 and Q2

    Thanks
    1. \vec{OP} = \vec{OA} + \vec{AP}
    =\vec{OA}+ \vec{OB}

    =\begin{pmatrix}2\\1\\-2\end{pmatrix}+\begin{pmatrix}2\\-1\\-1\end{pmatrix}

     =\begin{pmatrix}4\\0\\-3\end{pmatrix}
    So P is the point (4,0,-3).

    Find the coordinates of Q, R, S in a similar way.

    2. The plane passes through the origin, so its equation is of the form:
    ax + by + cz = 0
    We may divide through by a, to obtain an equation of the form:
    x + py + qz =0
    The point A lies in the plane. So:
    2+p-2q=0
    Similarly for B. Solve for p and q.

    3. Use the triple scalar product to find the volume of the parallelopiped:
    Volume = |\vec{OA}.(\vec{OB} \times \vec{OC})|
    which is most easily calculated using the 3 x 3 determinant form. For reference see here.

    Grandad
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  3. #3
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    Ohh i understand ! Thanks very much
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