The diagonals of a parallelogram are $3\hat{i} + \hat{j} - 2\hat{k}$ and $\hat{i} - 3\hat{j} + 4\hat{k}$. What is its area?

Explanation

1. 01

   $\vec{d_1} \times \vec{d_2} = -2\hat{i} - 14\hat{j} - 10\hat{k}$, of magnitude $\sqrt{4 + 196 + 100} = 10\sqrt{3}$.

2. 02

   From the diagonals, area $= \frac{1}{2}\lvert \vec{d_1} \times \vec{d_2} \rvert = 5\sqrt{3}$.