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Math Help - Finding a vertex of a tetrahedron given the volume (in Euclidian space).

  1. #1
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    Finding a vertex of a tetrahedron given the volume (in Euclidian space).

    So given that a particular pyramid (ABCD) in 3D euclidian space (with a triangular base) has a volume of 6 unitsł, how can you find the coordinates of a point F which lies on the line r= (t-1)i +(-t+1)j + (3t-3)k given that the volume of ABCF is the same as the volume of ABCD.

    for the record: A(1,2,3) B(2,3,1) C(3,1,2) D (6,6,6)


    help VERY appreciated !! thanks!!
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  2. #2
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    Quote Originally Posted by Yehia View Post
    So given that a particular pyramid (ABCD) in 3D euclidian space (with a triangular base) has a volume of 6 unitsł, how can you find the coordinates of a point F which lies on the line r= (t-1)i +(-t+1)j + (3t-3)k given that the volume of ABCF is the same as the volume of ABCD.

    for the record: A(1,2,3) B(2,3,1) C(3,1,2) D (6,6,6)


    help VERY appreciated !! thanks!!
    1. The volume of a terahedron with the edges \vec a, \vec b, \vec c is calculated by

    V=\frac16 (\vec a \times \vec b) \cdot \vec c

    2. With your example:

    V = \frac16 \left(\overrightarrow{AB} \times \overrightarrow{AC}  \right) \cdot \overrightarrow{AF}

    3. Since \overrightarrow{AF} = [t - 2, -t - 1, 3t - 6] you have to solve for t:

    \frac16 ([1, 1, -2] \times [2, -1, -1])\cdot [t - 2, -t - 1, 3t - 6] = 6

    4. Plug in the value of t into the equation of the line.
    Last edited by earboth; January 30th 2011 at 09:49 AM.
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