Question

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

• A. $\frac{1}{2}$
• B. $\frac{5}{2}$
• C. $\frac{3}{2}$
• D. 2

Hint:

Area of $\triangle$ with vectors $\vec{u}, \vec{v}$ is $\frac{1}{2}|\vec{u} \times \vec{v}|$.

Answer 1

The Correct Option is B

Step 1: Area Formula
Area of quadrilateral $ABCD$ = Area of $\triangle ABC$ + Area of $\triangle ADC$.
Area = $\frac{1}{2} |\overline{AB} \times \overline{AC}| + \frac{1}{2} |\overline{AC} \times \overline{AD}|$.

Step 2: Substitute Vectors
$\overline{AB} \times \overline{AC} = \overline{a} \times (2\overline{a} + 3\overline{b}) = 3(\overline{a} \times \overline{b})$.
$\overline{AC} \times \overline{AD} = (2\overline{a} + 3\overline{b}) \times \overline{b} = 2(\overline{a} \times \overline{b})$.

Step 3: Total Area
Total Area = $\frac{1}{2} [3|\overline{a} \times \overline{b}| + 2|\overline{a} \times \overline{b}|] = \frac{5}{2} |\overline{a} \times \overline{b}|$.

Step 4: Conclusion
Area of parallelogram = $|\overline{a} \times \overline{b}|$.
Thus, $\alpha = 5/2$.
Final Answer: (B)