Area of Triangle with Cross Product: Equation Variations

  • Thread starter Neen87
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  • #1
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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?
 

Answers and Replies

  • #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:
 
  • #3
LCKurtz
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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?
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.
 
  • #4
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Thanks so much! :-)
 

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