Why are areas sometimes negative?

Question

In coordinage geometry, sometimes calculating the area of a triangle, parallelogram, or any n-sided figure gives you a negative result. What is the reason behind this, and is there some known method to always get a positive result?

Answer 1

This is just the concept of orientation. For example: If you have two vectors $v = (v_1, v_2), w = (w_1, w_2) \in \mathbb R^2$ then the area of the parallelogram with vertices $0, v, v+w, w$ is given through $\det(v, w) = \det\begin{pmatrix} v_1 & w_1 \\ v_2 & w_2 \end{pmatrix}$. If you exchange $v$ and $w$ (remember, the determinant is multilinear and alternating), you get $\det(v, w) = -\det(w, v)$.

So just take the absolute value if you like to: $\lvert\det(v, w)\rvert = \lvert\det(w, v)\rvert$.