Change negative area of triangle into positive area!

Question

When calculating the area of a triangle (using determinants). What is the meaning of a negative determinant? Is it ok to just turn the value into a positive one?

Answer 1

When finding the area using determinants , negative determinant means that your $\hat i$ and $\hat j$ have swapped orientation . What you are interested in is the magnitude of the determinate, so you can take the absolute value of it.

EDIT :

you can also think of it this way, if the current determinant is

$det(A) = \begin{vmatrix}x_1&&y_1&&1\\x_2&&y_2&&1\\x_3&&y_3&&1\end{vmatrix}$ is negative, you can swap two rows to have the same magnitude of determinat but in opposite sign.

$det(\bar A) = \begin{vmatrix}x_2&&y_2&&1\\x_1&&y_1&&1\\x_3&&y_3&&1\end{vmatrix} = -det(A)$ which gives you the answer

Answer 2

Yes - determinant provides both the area and orientation (i.e. the handedness of the area). If you care only about the area, the sign can be ignored. But in other applications, the sign would matter.