Question

I : Area of quadrilateral $ABCD$ is $\frac{1}{2}|\overrightarrow{AC}\times \overrightarrow{BD}|$
II: Area of quadrilateral $ABCD$ is $|\overrightarrow{AB}\times \overrightarrow{AD}|$

• A. Only I is true
• B. Only II is true
• C. Both II and II true
• D. Neither l nor II are true

Answer 1

Answer: (A)

Only I is true

Let PV of $A,B,C,D$ be $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},\overrightarrow{d}$
Area of quad $=\frac{1}{2}|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{d}+\overrightarrow{d}\times \overrightarrow{a}|$
$=\frac{1}{2}|\overrightarrow{b}\times (\overrightarrow{c}-\overrightarrow{a})-\overrightarrow{d}\times (\overrightarrow{c}-\overrightarrow{a})|$
$=\frac{1}{2}|(\overrightarrow{c}-\overrightarrow{a})\times (\overrightarrow{b}-\overrightarrow{d})|$
$=\frac{1}{2}|\overrightarrow{AC}\times \overrightarrow{DB}|$