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Directional Derivative Solved Question: Explanation Needed Please

  1. Oct 23, 2012 #1
    1. Question:
    Find the directional derivatives of f(x, y, z) = x2+2xyz−yz2 at (1, 1, 2) in the directions parallel to the line (x−1)/2 = y − 1 = (z−2)/-3.



    2. Solution:
    We have ∇f = (2x + 2yz)i + (2xz - z2)j + (2xy - 2yz)k.

    Therefore, ∇f(1, 1, 2) = 6i - 2k.

    The given line is parallel to the vector v = (2, 1, -3).

    The corresponding unit vectors are u =  ± 1/||v|| and v = (±1/√14)(2, 1, -3).

    For the directional derivatives we find f'(1, 1, 2) = ∇f(1, 1, 2)dot(u) = ±18/√14



    3. My Questions:
    The only part of this that I have no clue about is how do they get the vector v from the information given.

    Could someone please explain how they find that direction vector?

    I get that since the lines are parallel the direction vector is the same but how does one find the directional vector?


    Any help is greatly appreciated.

    Thanks.
     
  2. jcsd
  3. Oct 23, 2012 #2

    tiny-tim

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    Hi Superdemongob! :smile:
    the (x-1) (y-1) and (z-2) are only there to make sure it goes through (1,1,2)

    if it went through (0,0,0) instead, the line would be x/2 = y/1 = z/-3,

    which is the vector v = (2, 1, -3) ! :smile:
     
  4. Oct 23, 2012 #3
    I get what you mean but how do you know that that is the directional vector?

    Like what is the method for finding out?

    If the question had the line 2(x−1) = y − 1 = -3(z−2) then how would the directional vector be different?



    Sorry but I'm really trying to understand this.
     
  5. Oct 23, 2012 #4

    tiny-tim

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    Hi Superdemongob! :smile:
    In that case, the line would still go through (1,1,-2),

    but the direction would be parallel to the direction 2x = y = -3z,

    to which the solution is (x,y,z) = (k/2,k,-k/3) for any value of k,

    ie the line containing the vector (1/2,1,-1/3) :wink:
     
  6. Oct 23, 2012 #5
    I think I finally get it.

    You take the inverse of the coefficients and that is your directional vector?
     
  7. Oct 23, 2012 #6

    tiny-tim

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    yes,

    and you can check it works because …

    it's the only answer that gives you 1/1 = 1/1 = 1/1 ! :biggrin:
     
  8. Oct 23, 2012 #7
    thank you so so much.

    i'm studying for a test and this helps a LOT.

    thanks again.
     
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