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Vectors in the plane.

  1. Aug 24, 2009 #1

    htk

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    can anyone please help me how to find the direction angle of a vector? Thank you!
     
  2. jcsd
  3. Aug 24, 2009 #2
    Think about the triangle that the vector and its components form.
     
  4. Aug 24, 2009 #3

    HallsofIvy

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    I presume you are talking about vectors in the plane since vectors in three dimensions have three "direction angles". How are you given the vector? If in x,y components, say ai+ bj, then b/a is the tangent of the angle the vector makes with the x-axis:
    [tex]\theta= arctan(\frac{a}{b})[/tex].
     
  5. Aug 24, 2009 #4

    tiny-tim

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    Welcome to PF!

    Hi htk! Welcome to PF! :smile:
    You find the cosine of the angle …

    which you do by finding the dot product. :wink:
     
  6. Aug 25, 2009 #5

    HallsofIvy

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    Re: Welcome to PF!

    His question was about a single vector. What do you want him to take the dot product with?
     
  7. Aug 25, 2009 #6

    tiny-tim

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    … just answering the question as asked! …

    with whatever mysterious entity he had in mind when he specified :wink:
     
  8. Aug 25, 2009 #7
    A two-dimensional vector [itex]\langle a,b \rangle[/itex] will have a direction angle [itex]\theta \text{ such that } \tan \theta = b / a[/itex] (not (a/b)) but this does not uniquely determine [itex]\theta[/itex], even if it is restricted to the interval [itex][0, 2 \pi )[/itex].

    You also need to consider in which quadrant does the vector lie. You need to adjust the value of [itex]\theta[/itex] so that it falls into the correct quadrant.

    For example, the vector [itex]\langle -3, 3 \rangle[/itex] has a direction angle so that [itex]\tan \theta = 3 / -3 = -1 \text{ which implies } \theta = -\pi /4 + n \pi[/itex] for an appropriate choince of integer n. Since the vector is in the second quadrant, we need to select the angle to fall there, so [itex]\theta = 3\pi / 4[/itex] (here n = 1).

    I hope this helps.

    --Elucidus
     
  9. Aug 25, 2009 #8
    Re: Welcome to PF!

    I suppose he could dot it with (1,0).

    EDIT: Modulo sign.
     
    Last edited: Aug 25, 2009
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