Equation of plane through point P containing line L

  1. 1. The problem statement, all variables and given/known data

    Find an equation for the plane W2 through the point P(2,3,-1) containing L

    [L = (3,-1,0) + t(5/3, -1/3, 1)]


    2. The attempt at a solution

    Let Q and R be two points on L where t = 3 and t = 6 respectively

    (t = 3) Q(11, -3, 3)
    (t = 6) R(16, -4, 6)

    Then PQ = (9, -6, 4) and PR = (14, -7, 7)

    The normal to the plane W2 is then

    PQ x PR = (14, -7, 21)

    Finally

    W2 : a(x - x0) + b(y - y0) + c(z - z0) = 0
    14(x - 2) - 7(y - 3) + 21(z - +1) = 0
    14x -7y +21z + 14 = 0

    Could someone please be so kind as to check my calculations (especially for the two points Q and R, I'm not sure if I derived them correctly from the line's equation)? I'm not sure if my answer is correct since I can't find any similar examples in my textbook so I'd just like to know if my reasoning is sound.

    Thanks in advance.

    phyz
     
  2. jcsd
  3. dynamicsolo

    dynamicsolo 1,662
    Homework Helper



    I understand why you chose t = 3 and t = 6 (though why not use t = 0?). But the values should be

    Q: (3,-1,0) + 3(5/3, -1/3, 1) = (3+5, -1-1, 0+3) = (8, -2, 3) and
    R: (3,-1,0) + 6(5/3, -1/3, 1) = (3+10, -1-2, 0+6) = (13, -3, 6) ,

    shouldn't they?

    Otherwise, in principle, your method looks all right...
     
    Last edited: Jun 18, 2008
  4. Thanks dynamicsolo! I just had a look here and I honestly can't tell you how I got to those values :confused:

    Just goes to show that you shouldn't attempt mathematics at the end of the day when the old grey matter is starting to rebel against work :biggrin:
     
    Last edited: Jun 19, 2008
  5. dynamicsolo

    dynamicsolo 1,662
    Homework Helper

    I tell the students I work with that mathematical reasoning engages many of the so-called "higher brain functions" (there is also a lot of interaction going on between various sections of the brain). Those functions are the first thing to go when you're tired or hungry, so you become prone to increasing numbers of mistakes (sometimes bizarre ones -- I've written stuff at 2 AM that made next to no sense after a night's sleep...).
     
  6. I am facing a similar problem right now, and i understand most of it, however, i am just having a hard time seeing how PQ * PR = normal of the plane
    Thank you!
     
  7. HallsofIvy

    HallsofIvy 40,242
    Staff Emeritus
    Science Advisor

    What do you mean by "PQ*PR"? It is the cross product of two vectors that is perpendicular to the plane containing the vectors and the standard notation for cross product is [itex]PQ \times PR[/itex]. And, of course, it is normal to the plane because the two vectors are in the plane and, as I said, "the cross product of two vectors is perpendicular to the plane containing them".
     
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