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Math Help - Finding the equation of a plane that contains two lines

  1. #1
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    Finding the equation of a plane that contains two lines

    I need help finding the equation a plane that contains the lines <t,2t,3t> and <3t,t,8t>
    I don't really know where to start. I tried setting the two lines equal to each other to find a point of intersection but it turned out funky. Any help would be appreciated.
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  2. #2
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    Re: Finding the equation of a plane that contains two lines

    I found the solution in case anyone else has the same problem.

    Lines intersect at origin O(0,0,0)
    r: [t,2t,3t] = t[1,2,3] = t u
    s: [3t,t,8t]=t[3,1,8] = t v

    where
    u = [1,2,3]
    v = [3,1,8]

    are the direction vectors of the lines

    a vector which is normal to the plane containing r and s is
    the cross product uxv

    u x v = det([i, j, k; 1, 2, 3; 3, 1, 8])

    u x v = 13i + j - 5k = [13,1,-5]

    so the plane containing r and s has equation

    13x + y - 5z = 0
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  3. #3
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    Re: Finding the equation of a plane that contains two lines

    Quote Originally Posted by jtl4231 View Post
    I need help finding the equation a plane that contains the lines <t,2t,3t> and <3t,t,8t>
    I don't really know where to start. I tried setting the two lines equal to each other to find a point of intersection but it turned out funky. Any help would be appreciated.
    It is clear that if t=0 then (0,0,0) is on both lines.

    Now the vector <1,2,3>\times<3,1,8>=<13,1,-5> is the normal to the plane.
    Thanks from jtl4231
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