Vector Algebra
Area of Parallelogram from Diagonal
nta_pyq_2024_jan
Grade 12
Question:
Let $A(2,3,5)$ and $C(-3,4,-2)$ be opposite vertices of a parallelogram $ABCD$. If the diagonal $\overrightarrow{BD}=\hat{i}+2\hat{j}+3\hat{k}$, then the area of the parallelogram is equal to:
$\dfrac{\sqrt{410}}{2}$
$\dfrac{\sqrt{474}}{2}$
$\dfrac{\sqrt{586}}{2}$
$\dfrac{\sqrt{306}}{2}$
Step-by-Step Solution
Key Concept: Area of parallelogram $=\dfrac{1}{2}|\overrightarrow{AC}\times\overrightarrow{BD}|$. Compute $\overrightarrow{AC}=C-A=(-5,1,-7)$.
$\overrightarrow{AC}=(-5,1,-7)$. $\overrightarrow{AC}\times\overrightarrow{BD}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\-5&1&-7\\1&2&3\end{vmatrix}=17\hat{i}+8\hat{j}-11\hat{k}$. Area $=\dfrac{1}{2}\sqrt{474}$.
Correct Answer: 2