Consider the three points $A=(2,3,4),B=(1,5,3),$ and $C=(2,7,4).$ What is the area of $\triangle ABC?$

Consider a triangle $\triangle OAB.$ If $\overrightarrow{OA}=(0,2,-2)$ and $\overrightarrow{OB}=(1,-4,1),$ then what is the area of $\triangle OAB?$

If $A=(0,1,1),B=(-1,2,3),$ and $C=(4,0,-1),$ what is the area of $\triangle ABC?$

Consider a parallelogram $ABCD.$ If $A=(-2,3,1),B=(0,5,3),$ and $D=(1,7,4),$ what is the square of the area of $ABCD?$

Calculate the area of the triangle spanned by the vectors $\vec{a}$ and $\vec{b},$ where $\lvert \vec{a} \rvert = 3 ,$$\lvert \vec{b} \rvert = 4$ and $\vec{a} \cdot \vec{b} = 8\sqrt{2}.$