The area of a triangle and determinants

[Yankel]

Dear all,

I was trying to prove that the area of a triangle is equal to the determinant consisting of the three points of the triangle. I got to the end, and something ain't working out. The signs are all wrong.

In the attached pictures I include my proof. Can you please tell me how can the two formulas be identical ? The first is the area coming from trapezoid subtraction , while the second is the determinant.

Thank you !

Clarification: when I say signs are opposite, I mean (y2-y3) vs. (y3-y2) , etc...

View attachment 7656

View attachment 7657

View attachment 7658

View attachment 7659

 

[I like Serena]

Hi Yankel,

The determinant gives double the area of a triangle that is oriented counter-clockwise.
In your case the triangle is oriented clockwise, meaning that we'll find the opposite.