Question

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

• A. $R(1+\sqrt{2})$
• B. $R(1+\sqrt{3})$
• C. $R(2+\sqrt{3})$
• D. $R(1+\sqrt{5})$

Answer 1

The correct option is A $R(1+\sqrt{2})$

$\text{Step 1:}$ $\text{Resolving Vector}$

Resolve vector $\overrightarrow{OB}$ in $x$ and $y$ direction.

$\text{[Ref. Fig. 2]}$

Magnitude of vector $\overrightarrow{OA},\overrightarrow{OB}$ and $\overrightarrow{OC}$ are same.

$|\overrightarrow{OA}|=|\overrightarrow{OB}|=|\overrightarrow{OC}|=R$

In $x$ direction $R_x=OA+OBcos{45}^o$

$R_x=R+\frac{R}{\sqrt{2}}$

In $y$ direction $R_y=OC+OBsin{45}^o$

$R_y=R+\frac{R}{\sqrt{2}}$

$\text{Step 2: Find resultant}$

Resultant $=\sqrt{R_x^2+R_y^2+2R_x{R}_{y}cos{90}^o}$

$=\sqrt{{\left[R(1+\frac{1}{\sqrt{2}})\right]}^2+{\left[R(1+\frac{1}{\sqrt{2}})\right]}^2}$

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

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

Hence option A is correct