Vector Algebra
Quadrilateral Area — Diagonal Cross Product
nta_pyq_2024_apr
Grade 12
Question:
If $A(3,1,-1)$, $B\left(\dfrac{5}{3},\dfrac{7}{3},\dfrac{1}{3}\right)$, $C(2,2,1)$ and $D\left(\dfrac{10}{3},\dfrac{2}{3},-\dfrac{1}{3}\right)$ are the vertices of a quadrilateral $ABCD$, then its area is
$\dfrac{2\sqrt{2}}{3}$
$\dfrac{5\sqrt{2}}{3}$
$2\sqrt{2}$
$\dfrac{4\sqrt{2}}{3}$
Step-by-Step Solution
Key Concept: Area $=\dfrac{1}{2}|\overrightarrow{BD}\times\overrightarrow{AC}|$. $\overrightarrow{BD}=\left(\dfrac{5}{3},-\dfrac{5}{3},-\dfrac{2}{3}\right)$. $\overrightarrow{AC}=(-1,1,2)$.
$\overrightarrow{BD}\times\overrightarrow{AC}=(-8/3,-8/3,0)$. Area $=4\sqrt{2}/3$.
Correct Answer: 4