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