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
- \( \frac{1}{2} \sqrt{410} \)
- \( \frac{1}{2} \sqrt{474} \)
- \( \frac{1}{2} \sqrt{586} \)
- \( \frac{1}{2} \sqrt{306} \)
The Correct Option is B
Solution and Explanation
The area is given by:
Area = \( \frac{1}{2} |\overrightarrow{AC} \times \overrightarrow{BD}| \)
Calculate \( \overrightarrow{AC} = (-5i + j - 7k) \) and \( \overrightarrow{BD} = i + 2j + 3k \) and find the cross product.
Then,
Area = \( \frac{1}{2} \sqrt{474} \)
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