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! :-)