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]

Welcome to PF!

Hi Neen87! Welcome to PF!

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

 

[LCKurtz]

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:

|v \times w| = |v||w|\sin\theta

where \theta 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 \theta or \pi - \theta. Either way you get the same value for its sine.

 

[Neen87]

Thanks so much! :-)