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Math Help - Altitudes of tetrahedron intersect, coordinate vectors

  1. #1
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    Altitudes of tetrahedron intersect, coordinate vectors

    A tetrahedron has vertices A, B, C and D with coordinate vectors for the points being:

    \mathbf{a}=\begin{pmatrix}0\\1\\2\end{pmatrix},\ \mathbf{b}=\begin{pmatrix}-1\\4\\1\end{pmatrix}, \mathbf{c}= \begin{pmatrix}1\\0\\3\end{pmatrix},\ and\ \mathbf{d}=\begin{pmatrix}-3\\1\\2\end{pmatrix}

    Find parametric vector equations for the two altitudes of the tetrahedron which pass through the vertices A and B, and determine whether the two altitudes intersect or not.

    Altitude is a line through the vertex and perpendicular to the opposite face.

    For my altitudes, I managed to get:

    \mathbf{x}=\begin{pmatrix}0\\1\\2\end{pmatrix}+\la  mbda\begin{pmatrix}-2\\6\\14\end{pmatrix} and\ \mathbf{x}=\begin{pmatrix}-1\\4\\1\end{pmatrix}+\lambda\begin{pmatrix}0\\-3\\3\end{pmatrix}

    Not too sure if that's right
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  2. #2
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    Quote Originally Posted by acevipa View Post
    A tetrahedron has vertices A, B, C and D with coordinate vectors for the points being:

    \mathbf{a}=\begin{pmatrix}0\\1\\2\end{pmatrix},\ \mathbf{b}=\begin{pmatrix}-1\\4\\1\end{pmatrix}, \mathbf{c}= \begin{pmatrix}1\\0\\3\end{pmatrix},\ and\ \mathbf{d}=\begin{pmatrix}-3\\1\\2\end{pmatrix}

    Find parametric vector equations for the two altitudes of the tetrahedron which pass through the vertices A and B, and determine whether the two altitudes intersect or not.

    Altitude is a line through the vertex and perpendicular to the opposite face.

    For my altitudes, I managed to get:

    \mathbf{x}=\begin{pmatrix}0\\1\\2\end{pmatrix}+\la  mbda\begin{pmatrix}-2\\6\\14\end{pmatrix} and\ \mathbf{x}=\begin{pmatrix}-1\\4\\1\end{pmatrix}+\lambda\begin{pmatrix}0\\-3\\3\end{pmatrix}

    Not too sure if that's right
    The equations of the lines containing the altitudes are correct.

    BTW: The two lines don't intersect.
    Last edited by earboth; June 3rd 2010 at 04:14 AM. Reason: Additional remark
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  3. #3
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    Quote Originally Posted by earboth View Post
    The equations of the lines containing the altitudes are correct.
    Ok great, so when it asks would they intersect, would you just equate the 2 lines and so the answer would be, no, they do not intersect?
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  4. #4
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    Quote Originally Posted by acevipa View Post
    Ok great, so when it asks would they intersect, would you just equate the 2 lines and so the answer would be, no, they do not intersect?
    Exactly! Nothing else.
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  5. #5
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    Quote Originally Posted by earboth View Post
    Exactly! Nothing else.
    Thanks mate
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