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Normal Plane to a Curve.

  1. Oct 27, 2009 #1
    1. The problem statement, all variables and given/known data

    At what point on the curve ⃗r(t) = (t^3, 3t, t^4) is the normal plane parallel to the plane 3x + 3y − 4z = 9 (the normal plane is the plane through the point ⃗r(t) which is normal to ⃗r′(t))

    2. Relevant equations

    I'm not really sure.

    3. The attempt at a solution
    (6t)(x-t^3) + (0)(y-3t) + (8t)(z-t^4) = 0

    But that got me nowhere.


    Thanks in advance.
     
  2. jcsd
  3. Oct 28, 2009 #2

    lanedance

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    i'm not sure what you attempted there...

    first find vector normal to the plane given, then find the tangent vector of the curve.. and have a think about how they will be related
     
  4. Oct 28, 2009 #3
    I read somewhere that that would be the equation of a normal plane to a curve. But it didn't work.

    I did what you said, and they'll be related in that they'll be parallel vectors. I tried doing what you said and setting them equal to each other, but I just got equations for t that seem insolvable. ( for instance, 0= 8t^4 + 9 +16t^6)
     
  5. Oct 28, 2009 #4

    lanedance

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    that's not what i get, it works out ok... i'm not sure how you get the higher powers of t in your equation either

    what do you get for the normal to the plane & for the tangent vector?
     
  6. Oct 28, 2009 #5
    The normal is the gradient, so I got (3,3,-4). And I got (3t^2,3,4t^3)/sqrt(9t^4+9+16t^6) for the tangent vector.
     
  7. Oct 28, 2009 #6

    lanedance

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    both look good, but I see you are normalising the tangent vector to length 1, that's not needed here, as you just need to know when its parallel

    so if p is normal to the plane, t is the tangent, you just need to know when t = c.p for any constant c, which shows they are parallel. This should lead to a reasonably easy equation set if you don't normalise the vector
     
    Last edited: Oct 28, 2009
  8. Oct 28, 2009 #7
    Well alright then, lol. Thanks guys, got it all figured out now. The answer is (-1,-3,1). Or rather, I'm assuming that that's the correct answer because it's what I got and it matches up with one of the multiple choice options :P
     
  9. Oct 28, 2009 #8

    lanedance

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    yep thats what i get
     
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