Three vectors \(\overrightarrow{\mathrm{OP}}, \overrightarrow{\mathrm{OQ}}\) and \(\overrightarrow{\mathrm{OR}}\) 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 _______.

Explanation:
1. Components of the Vectors: Based on the given coordinate system in the figure, we can resolve each vector into its horizontal (\( \hat{i} \)) and vertical (\( \hat{j} \)) components:
• \( \vec{\text{OP}} \) acts along the positive x-axis: \( \vec{\text{OP}} = A\hat{i} \)
• \( \vec{\text{OQ}} \) acts along the positive y-axis: \( \vec{\text{OQ}} = A\hat{j} \)
• \( \vec{\text{OR}} \) acts at an angle of \( 45^\circ \) below the positive x-axis (in the fourth quadrant): \( \vec{\text{OR}} = A\cos 45^\circ\hat{i} - A\sin 45^\circ\hat{j} = \frac{A}{\sqrt{2}}\hat{i} - \frac{A}{\sqrt{2}}\hat{j} \)

2. Finding the Resultant Vector (\( \vec{R} \)):
Summing all three vectors together:
\( \vec{R} = \vec{\text{OP}} + \vec{\text{OQ}} + \vec{\text{OR}} \)
\( \vec{R} = (A\hat{i}) + (A\hat{j}) + \left(\frac{A}{\sqrt{2}}\hat{i} - \frac{A}{\sqrt{2}}\hat{j}\right) \)
\( \vec{R} = \left(A + \frac{A}{\sqrt{2}}\right)\hat{i} + \left(A - \frac{A}{\sqrt{2}}\right)\hat{j} \)

3. Calculating the Magnitude (\( |\vec{R}| \)):
\( |\vec{R}| = \sqrt{\left(A + \frac{A}{\sqrt{2}}\right)^2 + \left(A - \frac{A}{\sqrt{2}}\right)^2} \)
\( |\vec{R}| = \sqrt{\left(A^2 + \sqrt{2}A^2 + \frac{A^2}{2}\right) + \left(A^2 - \sqrt{2}A^2 + \frac{A^2}{2}\right)} \)
\( |\vec{R}| = \sqrt{2A^2 + A^2} = \sqrt{3A^2} = A\sqrt{3} \)

Comparing this with the given resultant magnitude \( A\sqrt{x} \):
\( A\sqrt{x} = A\sqrt{3} \implies x = 3 \)

Hence, Option D is the correct choice.