How to calculate the area of a triangle ABC when given three position vectors $a, b$, and $ c$ in 3D?

Heron works of course but it would be simpler to take half the length of the cross product $(b-a)\times(c-a)$.

Solution: Construct the vectors $\hat{ab}$, $\hat{ac}$ and take $\frac{1}{2} |\hat{ab} \times \hat{ac}|$. We take half of the resulting since the original gives the area of the parallelogram decsribed by the vectors.

use this formula: $$S=\sqrt{p(p-s_1)(p-s_2)(p-s_3)}$$ where $p=\frac{s_1+s_2+s_3}{2}$, and $s_1,s_2,s_3$ are the lengths of the three sides. You can get $s_1,s_2,s_3$ from the three positions.