Solved! Cross product in parallelogram

Question

The area of the parallelogram $ABCD$ is equal to: $$\text{Area} = \left|\vec{AB}\times\vec{AD}\right|\text.$$ Is it the same for $$\text{Area} = \left|\vec{AC}\times\vec{AD}\right|\text?$$ Because both will derive into the height and the base. If this is true, can we somehow prove: $$\left|\vec{AB}\times\vec{AD}\right| = \left|\left(\vec{AB}+\vec{AD}\right)\times\vec{AD}\right|$$

Thanks!

Parallelogram

Answer 1

Indeed, using properties of the cross product, $$ \left|\left(\vec{AB}+\vec{AD}\right)\times\vec{AD}\right|= \left|\vec{AB}\times\vec{AD}+\vec{AD}\times\vec{AD}\right |=$$

$$\left| \vec{AB}\times\vec{AD}+\vec{0} \right |=\left|\vec{AB}\times\vec{AD}\right| . $$