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Find equation of a plane containing two lines

  1. Oct 8, 2009 #1
    1. The problem statement, all variables and given/known data
    find the equation of the plane that contains the line r_1(t) = (t,2t,3t) and r_2(t)=(3t,t,8t)


    2. Relevant equations



    3. The attempt at a solution
    i don't know where to start...my book does not have an example similar. can somebody just point me to the right direction?
     
  2. jcsd
  3. Oct 8, 2009 #2

    Mark44

    Staff: Mentor

    Each of your line equations defines a vector with the same direction as the line. Find the cross product of these vectors to get a third vector, say <a, b, c>. That will be a normal to the plane. Find a point on either of your given lines, say (x0, y0, z0).

    Use the normal and the given point to write the equation of the plane as a(x - x0) + b(y - y0) + c(z - z0) = 0.
     
  4. Oct 8, 2009 #3
    [tex]r_1=t(1,2,3)[/tex]

    [tex]r_2=t(3,1,8)[/tex]

    Both [tex]\vec{v}_1=\left(\begin{array}{c} 1 \\ 2 \\ 3\end{array}\right)[/tex] and [tex]\vec{v}_2=\left(\begin{array}{c} 3 \\ 1 \\ 8\end{array}\right)[/tex] are in the plane.

    So [tex]\vec{n}=\vec{v}_1 \times \vec{v}_2=\left(\begin{array}{c} 13 \\ 1 \\ -5\end{array}\right)[/tex] is normal to the plane.

    Hence the equation:

    [tex]13x+y-5z=0[/tex]
     
  5. Oct 8, 2009 #4

    Mark44

    Staff: Mentor

    Donaldos,
    It is the policy of this forum to provide help to a poster, but not to give a complete answer to someone's problem.
     
  6. Oct 8, 2009 #5
    ok i got the answer, but when i did the cross product my answers sitll have the t in them

    my ans is 13t2x+t2y-5t2z = 0

    the books answer is exactly that but without the t's
     
  7. Oct 8, 2009 #6
    I'm sorry. I'll keep that in mind.
     
  8. Oct 8, 2009 #7
    i see donaldos wrote r with the t outside before doing the determinant...what happens to that?
     
  9. Oct 8, 2009 #8

    Mark44

    Staff: Mentor

    All you need is any old vector that is parallel to the line, so any multiple of the vector will still be parallel. The t multiplier can be any real value, so it's convenient to let t = 1.
     
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