JEE Main · 2024 · 29 Jan 2024 · Shift 2 · Vector Algebra

Take $O$ as origin so the position vectors of $A,B,C$ are $\vec{a},12\vec{a}+4\vec{b},\vec{b}$.

For a polygon with position vectors $P_0,P_1,\dots$, the (signed) area is $\frac{1}{2}\left|\sum P_i\times P_{i+1}\right|$. For $O,A,B,C$:

$\overrightarrow{OA}\times \overrightarrow{AB}$-type sum reduces to

$O\times A+A\times B+B\times C+C\times O=0+\vec{a}\times (12\vec{a}+4\vec{b})+(12\vec{a}+4\vec{b})\times \vec{b}+0$

$=4(\vec{a}\times \vec{b})+12(\vec{a}\times \vec{b})=16(\vec{a}\times \vec{b})$.

Area of $OABC=\frac{1}{2}\cdot 16|\vec{a}\times \vec{b}|=8|\vec{a}\times \vec{b}|$.

Area of $S$ (parallelogram on $OA,OC$) $=|\vec{a}\times \vec{b}|$.

Ratio $=8$.