JEE Main 2024 (Online) 29th January Evening ShiftQuestion 1
Let $\overrightarrow{OA}=\vec a$, $\overrightarrow{OB}=12\vec a+4\vec b$ and $\overrightarrow{OC}=\vec b$, where $O$ is the origin. If $S$ is the parallelogram with adjacent sides $OA$ and $OC$, then $\frac{\text{area of the quadrilateral }OABC}{\text{area of }S}$ is equal to
Vector Algebra
Solution
Area of $OABC=\frac12|\overrightarrow{OB}\times\overrightarrow{AC}|=\frac12|(12\vec a+4\vec b)\times(\vec b-\vec a)|=\frac12\cdot16|\vec a\times\vec b|=8|\vec a\times\vec b|$, and area of $S=|\vec a\times\vec b|$, so the ratio is $8$.