Vector Algebra Question 10
Question 10 - 2024 (30 Jan Shift 1)
Let $A(2,3,5)$ and $C(-3,4,-2)$ be opposite vertices of a parallelogram $\mathrm{ABCD}$ if the diagonal $\overrightarrow{B D}=\hat{i}+2 \hat{j}+3 \hat{k}$ then the area of the parallelogram is equal to
(1) $\frac{1}{2} \sqrt{410}$
(2) $\frac{1}{2} \sqrt{474}$
(3) $\frac{1}{2} \sqrt{586}$
(4) $\frac{1}{2} \sqrt{306}$
Show Answer
Answer (2)
Solution
Area $=|\overrightarrow{\mathrm{AC}} \times \overrightarrow{\mathrm{BD}}|$
$=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \ 5 & -1 & 7 \ 1 & 2 & 3\end{array}\right|$
$=\frac{1}{2}|-17 \hat{i}-8 \hat{j}+11 \hat{k}|=\frac{1}{2} \sqrt{474}$
