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Math Help - lines in space

  1. #1
    Junior Member
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    lines in space

    Hi, Im trying to find a parametric equation of a line through (2,3,0) and perpendicular to vectors u = i+2j+3k and v = 3i+4j+5k?
    To do this I think i need a parallel vector, how do i get this? is it something to do with the cross product?
    thanks.
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  2. #2
    MHF Contributor

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    Both of your ideas are good, just go ahead: \vec{u}\times\vec{v} is a vector orthogonal to both \vec{u} and \vec{v}. Since the orthogonal complement of a plane in \mathbb{R}^3 is a line, \vec{u}\times\vec{v} is a directing vector for the line you want. So a parametric equation for this line is: \lambda\mapsto (2,3,0)+\lambda \vec{u}\times\vec{v}, defined for \lambda\in\mathbb{R}.
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