Question

$ABCD$ is a quadrilateral with $\overrightarrow{AB}=\vec{a},\overrightarrow{AD}=\vec{b}$ and $\overrightarrow{AC}=2\vec{a}+3\vec{b}$. If its area is $\alpha$ times the area of the parallelogram $gm$ with $AB,AD$ as adjacent sides, then find the value of $\alpha$.

Answer 1

Answer: 2.50

Area of quadrilateral $ABCD$
= area of $△ABC+$ area of $△ACD$
$=\frac{1}{2}[\vec{a}\times (2\vec{a}+3\vec{b})]+\frac{1}{2}[(2\vec{a}+3\vec{b})\times \vec{b}]$
$=\frac{1}{2}(3\vec{a}\times \vec{b}+2\vec{a}\times \vec{b})=\frac{5}{2}(\vec{a}\times \vec{b})$
$=\frac{5}{2}$ area of parallelogram $ABCD$
$\Rightarrow \alpha =\frac{5}{2}$