Q 11-03-022JEE MainMedium
The diagonals of a parallelogram are $\vec{d}_1 = 3\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{d}_2 = \hat{i} - 3\hat{j} + 4\hat{k}$. The area of the parallelogram is
Answer: (C) $5\sqrt{3}$
Concept. Area of a parallelogram from its diagonals: $\dfrac12|\vec d_1 \times \vec d_2|$.
Step 1. Cross product.
$$\vec d_1 \times \vec d_2 = \begin{vmatrix} \hat i & \hat j & \hat k \\ 3 & 1 & -2 \\ 1 & -3 & 4 \end{vmatrix} = -2\hat i - 14\hat j - 10\hat k$$
Step 2. Area. $|\vec d_1 \times \vec d_2| = \sqrt{300} = 10\sqrt3$, so area $= 5\sqrt3$.
Answer. $5\sqrt3$ (Option C).
Solution by Sreeraj P, M.Sc Physics