Question

The diagonals of a parallelogram are the vectors \(3\hat{i}+6\hat{j}-2\hat{k}\) and \(-\hat{i}-2\hat{j}-8\hat{k}\) then the length of the shorter side of parallelogram is

• A. \(2\sqrt3\)
• B. \(\sqrt{14}\)
• C. \(3\sqrt5\)
• D. \(4\sqrt3\)

Answer 1

The Correct Option is B

Let the diagonals of the parallelogram be represented as vectors \( \mathbf{d_1} \) and \( \mathbf{d_2} \): \[ \mathbf{d_1} = 3 \hat{i} + 6 \hat{j} - 2 \hat{k}, \quad \mathbf{d_2} = - \hat{i} - 2 \hat{j} - 8 \hat{k} \] The length of a diagonal in a parallelogram is given by the magnitude of the vector representing that diagonal: \[ |\mathbf{d_1}| = \sqrt{3^2 + 6^2 + (-2)^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \] \[ |\mathbf{d_2}| = \sqrt{(-1)^2 + (-2)^2 + (-8)^2} = \sqrt{1 + 4 + 64} = \sqrt{69} \] Now, let the side vectors of the parallelogram be \( \mathbf{a} \) and \( \mathbf{b} \). The relationship between the diagonals and the sides of the parallelogram is given by the following formulas: \[ \mathbf{d_1} = \mathbf{a} + \mathbf{b}, \quad \mathbf{d_2} = \mathbf{a} - \mathbf{b} \] The length of the shorter side can be obtained using the following equation: \[ |\mathbf{a}| = \sqrt{\frac{|\mathbf{d_1}|^2 + |\mathbf{d_2}|^2}{2}} = \sqrt{\frac{7^2 + (\sqrt{69})^2}{2}} = \sqrt{\frac{49 + 69}{2}} = \sqrt{59} \] Thus, the length of the shorter side is \( \sqrt{14} \).

Thus, the correct answer is (B) : \(\sqrt{14}\).