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Thread: Vector: Lines Intersect

  1. #1
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    Vector: Lines Intersect

    Vectors have to be my weakest points. How would I do this question, will I need to find $\displaystyle s$ and $\displaystyle t$? Help would be appreciated.

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    Q:

    Referred to a fixed origin $\displaystyle O$, the lines $\displaystyle l_1$ and $\displaystyle l_2$ have equations $\displaystyle \bold r = 3 \bold i + 6 \bold j + \bold k + s (2 \bold i + 3 \bold j - \bold k)$ and $\displaystyle \bold r = 3 \bold i - \bold j + 4 \bold k + t ( \bold i -2 \bold j + \bold k) $ respectively, where $\displaystyle s$ and $\displaystyle t$ are scalar parameters.

    Show that $\displaystyle l_1$ and $\displaystyle l_2$ intersect and determine the position vector of their point of intersection.


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    Thanks in advance.
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  2. #2
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    Rewrite the lines as: $\displaystyle l_1 = \left\{ \begin{array}{r}
    3 + 2s \\
    6 + 3s \\
    1 - s \\
    \end{array} \right.\quad ,\quad l_2 = \left\{ \begin{array}{r}
    3 + t \\
    - 1 - 2t \\
    4 + t \\
    \end{array} \right.$
    Now just solve the system.
    But make sure the the solution works for all three equations.
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