I know how to find the area of a triangle in three-space, say ABC: First I would find $\displaystyle \vec{AB}$ and then $\displaystyle \vec{AC}$ and then use the formula for the area of a parallelogram:

$\displaystyle A = {\vec{AB} \times \vec{AC}$

Finding the magnitude of that and dividing it by two (since the area of a triangle is half that of the area of a parallelogram), I would find the area of the triangle ABC.

But, how would I find the area of a triangle that does not have a Z component (is in two-space)? For example: Find the area of the triangle with vertices of (1, -2), (-1, 3), (2, 4). Since this is in two-space, I can't take the cross product of any two of these vectors to find the area. According to my book, the area of this triangle is $\displaystyle \frac{17}{2}$ and I have no idea how they computed that. Any help? I would appreciate learning a systematic approach to solving the areas of a triangle in two-space. Thanks in advance.