Area of Triangle with Cross Product: Equation Variations

Neen87

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?

tiny-tim

Hi Neen87! Welcome to PF!

Hint: call the vertices a b and c, so the sides are a - b etc.

LCKurtz

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.

Neen87

Thanks so much! :-)