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Area of Triangle with Cross Product: Equation Variations

  1. Dec 31, 2009 #1
    Hello!

    I'm trying to understand how this formula: 1/2 magnitude of v × w (area of triangle)
    yields the same value no matter which 2 adjacent sides are chosen.

    How would you prove mathematically that this is the case?
     
  2. jcsd
  3. Dec 31, 2009 #2

    tiny-tim

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    Welcome to PF!

    Hi Neen87! Welcome to PF! :smile:

    Hint: call the vertices a b and c, so the sides are a - b etc. :wink:
     
  4. Dec 31, 2009 #3

    LCKurtz

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    Because if you draw the parallelogram with sides v and w, the cross product magnitude gives:

    [tex]|v \times w| = |v||w|\sin\theta[/tex]

    where [itex]\theta[/itex] is the angle between the two vectors you have chosen for sides. Now, whichever two sides you choose and whichever direction they point, the angle between them will be either [itex]\theta[/itex] or [itex]\pi - \theta[/itex]. Either way you get the same value for its sine.
     
  5. Dec 31, 2009 #4
    Thanks so much! :-)
     
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