Question

The area of a parallelogram with \( 3\hat{i} + \hat{j} - 2\hat{k} \) and \( \hat{i} - 3\hat{j} + 4\hat{k} \) as diagonals is

• A. \( \sqrt{72} \)
• B. \( \sqrt{73} \)
• C. \( \sqrt{74} \)
• D. \( \sqrt{75} \)

Hint:

The area of a parallelogram can be found using the magnitude of the cross product of the vectors representing its sides.

Answer 1

The Correct Option is D

Step 1: Formula for the area of a parallelogram.
The area of a parallelogram formed by vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by the magnitude of the cross product: \[ \text{Area} = |\mathbf{A} \times \mathbf{B}| \] Here, \( \mathbf{A} = 3\hat{i} + \hat{j} - 2\hat{k} \) and \( \mathbf{B} = \hat{i} - 3\hat{j} + 4\hat{k} \). The cross product yields the area \( \sqrt{75} \).

Step 2: Conclusion.
Thus, the correct answer is option (D).

Final Answer: \[ \boxed{\text{(D) } \sqrt{75}} \]