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(B) Let the vectors be represented in a Cartesian coordinate system with $O$ at the origin $(0,0)$.Vector $\overrightarrow{OA}$ is along the positive

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$R(1 + \sqrt{2})$

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(B) Let the vectors be represented in a Cartesian coordinate system with $O$ at the origin $(0,0)$.Vector $\overrightarrow{OA}$ is along the positive $x$-axis: $\overrightarrow{OA} = R\hat{i}$.Vector $\overrightarrow{OC}$ is along the positive $y$-axis: $\overrightarrow{OC} = R\hat{j}$.Vector $\overrightarrow{OB}$ makes an angle of $45^{\circ}$ with the $x$-axis: $\overrightarrow{OB} = R\cos(45^{\circ})\hat{i} + R\sin(45^{\circ})\hat{j} = \frac{R}{\sqrt{2}}\hat{i} + \frac{R}{\sqrt{2}}\hat{j}$.The resultant vector $\vec{R}_{net} = \overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC} = (R + \frac{R}{\sqrt{2}})\hat{i} + (R + \frac{R}{\sqrt{2}})\hat{j}$.The magnitude is $|\vec{R}_{net}| = \sqrt{(R + \frac{R}{\sqrt{2}})^2 + (R + \frac{R}{\sqrt{2}})^2} = \sqrt{2(R + \frac{R}{\sqrt{2}})^2} = \sqrt{2}(R + \frac{R}{\sqrt{2}}) = R\sqrt{2} + R = R(\sqrt{2} + 1)$.

Find the resultant of three vectors $\overrightarrow{OA}$,$\overrightarrow{OB}$,and $\overrightarrow{OC}$ as shown in the figure. The radius of the circle is $R$.

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