Area of a triangle in 3 space using cross product

[jlemus85]

Hi all, I have a general question.

When calculating the area of a triangle in 3 space, one applies the formula

1/2||axb||. Given three vertices, a,b,c...does it matter which vectors we choose to use (ab, bc, ac) as our a b vectors?

Thanks!

 

[HallsofIvy]

First, because of the norm, it does not matter which is a and which is b: that is (1/2)||axb||= (1/2)||bxa||. That is, order doesn't matter. Now if a, b, and c are vectors forming a triangle, then c= a+ b so (1/2)||axc||= (1/2)||ax(a+b||= (1/2)||axa+ ab|= (1/2)||axb|| because axa= 0. In other words, all combinations give the same result.