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Math Help - Find parametric equations for the line

  1. #1
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    Question Find parametric equations for the line

    Question
    Lsub1 : x=1+2t, y=2-t, z=4-2t
    Lsub2 : x=9+t , y=5+3t, z=-4-t

    Lsub1 and Lsub2 intersect at the point (7,-1,-2).

    Find parametric equations for the line that is perpendicular to Lsub1 and Lsub2 passes through their point of intersection.

    * if possible, please draw me the picture. I can't figure out how this line can perpendicular to Lsub1 and Lsub2.

    Thank you very much.
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  2. #2
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    Quote Originally Posted by Jenny20 View Post
    Question
    Lsub1 : x=1+2t, y=2-t, z=4-2t
    Lsub2 : x=9+t , y=5+3t, z=-4-t

    Lsub1 and Lsub2 intersect at the point (7,-1,-2).

    Find parametric equations for the line that is perpendicular to Lsub1 and Lsub2 passes through their point of intersection.

    * if possible, please draw me the picture. I can't figure out how this line can perpendicular to Lsub1 and Lsub2.

    Thank you very much.
    Consider the xyz plane.
    The x-axis and y-axis intersect.
    And z-axis is perpendicular to both of them.
    That is how the visuallization looks.

    Line 1) Is algined with the vector,
    \bold{u}=2\bold{i}-\bold{j}-2\bold{k}

    Line 2) Is algined with vector,
    \bold{v}=\bold{i}+3\bold{j}-\bold{k}

    The line that we seek is perpendicular to both, that is a vector that is orthogonal to both these non-zero vectors.

    We can use the cross product,
    \bold{u}\times \bold{v} =\left| \begin{array}{ccc}\bold{i}&\bold{j}&\bold{k} \\ 2&-1&-2\\ 1&3&-1 \end{array} \right| =7\bold{i}+7\bold{k}.

    We can remove the 7 multiple, because if it is algined with <7,0,7> it is certainly aligned with <1,0,1>.

    Thus, the parametric equations are,
    \left\{ \begin{array}{c} x=7+t\\y=-1\\z=-2+t \end{array} \right\}
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