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Math Help - Line intersection

  1. #1
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    Line intersection

    Let Line x be defined by  \bar {p_{1}}(t) = \bar {p_{1}}+t \bar {V_{1}} and
    let Line y be defined by  \bar {p_{2} }(s) = \bar {p_{2}} + s \bar {V_{2}}

    Find conditions on line x,y such that they have unique point of intersection. Find this point.

    Would that be  \bar {V_{1}} \dot \bar {V_{2}} \neq 0 ? And how would I find the point?
    Last edited by tttcomrader; January 27th 2008 at 08:59 AM.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Would that be  \bar {V_{1}} \dot \bar {V_{2}} \neq 0 ?
    If you're working in two dimensions, yes. If you're working in more dimensions, you need to make sure they're not skew; the easiest way to do that is to attempt to find your intersection.

    And how would I find the point?
    Set the two equations equal to each other and solve for t, then plug t into both your equations to make sure it gives you the same point in each. If it does, you've found your intersection. If it doesn't, the lines are skew.
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  3. #3
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    Quote Originally Posted by tttcomrader View Post
    Let Line x be defined by  \bar {p_{1}}(t) = \bar {p_{1}}+t \bar {V_{1}} and
    let Line y be defined by  \bar {p_{2} }(s) = \bar {p_{2}} + s \bar {V_{2}}

    Find conditions on line x,y such that they have unique point of intersection. Find this point.

    Would that be  \bar {V_{1}} \dot \bar {V_{2}} \neq 0 ? And how would I find the point?
    The 2 lines intersect even in 3-D if
    <br />
(\overrightarrow{p_1} - \overrightarrow{p_2}) \cdot \frac{\overrightarrow{V_1} \times \overrightarrow{V_2}}{| \overrightarrow{V_1} \times \overrightarrow{V_2} |} = 0

    With this formula you calculate the distance between 2 lines. If the distance is zero then there must be a point of intersection.
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