Question

If $ \hat{i} + \hat{j} - \hat{k} $ and $ 2\hat{i} - 3\hat{j} + \hat{k} $ are adjacent sides of a parallelogram, then length of its diagonal is

• A. \( \sqrt{3}, \sqrt{14} \)
• B. \( \sqrt{13}, \sqrt{14} \)
• C. \( \sqrt{21}, \sqrt{3} \)
• D. \( \sqrt{21}, \sqrt{13} \)

Hint:

To calculate the length of diagonals of a parallelogram, use the formula for the magnitudes of \( \vec{a} + \vec{b} \) and \( \vec{a} - \vec{b} \), where \( \vec{a} \) and \( \vec{b} \) are the vectors representing the adjacent sides.

Answer 1

The Correct Option is D

The length of the diagonal of a parallelogram with adjacent sides \( \vec{a} \) and \( \vec{b} \) is given by the formula: \[ |\vec{a} + \vec{b}| \quad \text{and} \quad |\vec{a} - \vec{b}| \] Where: - \( \vec{a} = \hat{i} + \hat{j} - \hat{k} \) - \( \vec{b} = 2\hat{i} - 3\hat{j} + \hat{k} \) Now calculate \( \vec{a} + \vec{b} \) and \( \vec{a} - \vec{b} \): \[ \vec{a} + \vec{b} = (\hat{i} + \hat{j} - \hat{k}) + (2\hat{i} - 3\hat{j} + \hat{k}) = 3\hat{i} - 2\hat{j} \] \[ \vec{a} - \vec{b} = (\hat{i} + \hat{j} - \hat{k}) - (2\hat{i} - 3\hat{j} + \hat{k}) = -\hat{i} + 4\hat{j} - 2\hat{k} \] Now find the magnitudes of these vectors: \[ |\vec{a} + \vec{b}| = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \] \[ |\vec{a} - \vec{b}| = \sqrt{(-1)^2 + 4^2 + (-2)^2} = \sqrt{1 + 16 + 4} = \sqrt{21} \] Thus, the length of the diagonals are \( \sqrt{21} \) and \( \sqrt{13} \).