If $A$, $B$ and $C$ are three non-collinear points in a plane such that their relative position vectors form the directed sides $\vec{AB} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{AC} = 2\hat{i} - 3\hat{j}$, then compute the exact geometric area of the triangle $\Delta ABC$.
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When expanding the cross product determinant, remember that the middle term ($\hat{j}$) always carries a negative sign by default from matrix cofactor rules: $+\hat{i}, -\hat{j}, +\hat{k}$.