Question

Three vectors \(\overrightarrow{OP}\), \(\overrightarrow{OQ}\), and \(\overrightarrow{OR}\) each of magnitude \(A\) are acting as shown in figure. The resultant of the three vectors is \(A \sqrt{x}\). The value of \(x\) is ______.

Answer 1

Correct Answer: 3

To determine the value of \( x \) for which the resultant of vectors \(\overrightarrow{OP}\), \(\overrightarrow{OQ}\), and \(\overrightarrow{OR}\) equals \( A \sqrt{x} \), analyze the vector setup. Given \(\angle POQ = 90^\circ\) and \(\angle QOR = 135^\circ\), use vector addition. The vectors are defined as: \[ \overrightarrow{OP} = A \hat{i}, \quad \overrightarrow{OQ} = A \hat{j}, \quad \overrightarrow{OR} = -A \cos(45^\circ) \hat{i} - A \sin(45^\circ) \hat{j} \] This simplifies to: \[ \overrightarrow{OR} = -A \frac{\sqrt{2}}{2} \hat{i} - A \frac{\sqrt{2}}{2} \hat{j} \] Summing the vectors yields: \[ \overrightarrow{OP} + \overrightarrow{OQ} + \overrightarrow{OR} = \left(A - A \frac{\sqrt{2}}{2}\right) \hat{i} + \left(A - A \frac{\sqrt{2}}{2}\right) \hat{j} \] The magnitude of the resultant vector \(\overrightarrow{R}\) is calculated as: \[ |\overrightarrow{R}| = \sqrt{\left(A - A \frac{\sqrt{2}}{2}\right)^2 + \left(A - A \frac{\sqrt{2}}{2}\right)^2} \] \[ = \sqrt{2 \left(A - A \frac{\sqrt{2}}{2}\right)^2} \] \[ = \sqrt{2 \left(A^2 \left(1-\frac{\sqrt{2}}{2}\right)^2\right)} \] \[ = A \sqrt{2 \left(1 - \frac{\sqrt{2}}{2}\right)^2} \] Equating the magnitude to \( A \sqrt{x} \) gives: \[ A \sqrt{x}= A \sqrt{2 \left(1 - \frac{\sqrt{2}}{2}\right)^2} \] Therefore, \[ x = 2 \left(1 - \frac{\sqrt{2}}{2}\right)^2 \] Performing the calculation: \[ 1 - \frac{\sqrt{2}}{2} = \frac{2-\sqrt{2}}{2} \] \[ \left(\frac{2-\sqrt{2}}{2}\right)^2 = \frac{(2-\sqrt{2})^2}{4} = \frac{4 - 4\sqrt{2} + 2}{4} = \frac{6 - 4\sqrt{2}}{4} = \frac{3 - 2\sqrt{2}}{2} \] Substituting back into the equation for \( x \): \[ x = 2 \times \left(\frac{3-2\sqrt{2}}{2}\right) \] \[ x = 3 \] The value of \( x \) is 3.