Three vectors $$\vec{OP}$$, $$\vec{OQ}$$ and $$\vec{OR}$$ 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 ________.

Let's write the three vectors given in their deconstructed form:

$$\vec{OP} = A\hat{i}$$

$$\vec{OQ} = A\hat{j}$$

$$\vec{OR} = \dfrac{A}{\sqrt{2}}\hat{i} - \dfrac{A}{\sqrt{2}}\hat{j}$$

Thus, adding the vectors, we get,

$$\vec{OP}+\vec{OQ}+\vec{OR} =\vec{R} =  \left(A + \dfrac{A}{\sqrt{2}}\right)\hat{i} + \left(A - \dfrac{A}{\sqrt{2}}\right)\hat{j}$$

And hence, the magnitude of the resultant vector,

$$|\vec{R}| = \sqrt{\left(A + \dfrac{A}{\sqrt{2}}\right)^2 + \left(A - \dfrac{A}{\sqrt{2}}\right)^2}$$

$$|\vec{R}| = \sqrt{2A^2 + 2\dfrac{A^2}{2}} = A\sqrt{3}$$

Therefore, the value of $$x$$ is $$3$$.