Question

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

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

Answer 1

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

$\text{Step 1: Resolving Vector}$$\text{[Ref. Fig. 2]}$

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

$\text{Step 2: Resultant of vectors in x and y direction[Ref. Fig. 3]}$

Magnitude of vector $\overrightarrow{OA},\overrightarrow{OB}$ and $\overrightarrow{OC}$ are same and equal to $R$

In $x$ direction $R_x=\overrightarrow{OA}+\overrightarrow{OB}\cos {45}^o=R+\frac{R}{\sqrt{2}}$

In $y$ direction $R_y=\overrightarrow{OC}+\overrightarrow{OB}\sin {45}^o=R+\frac{R}{\sqrt{2}}$

$\text{Step 3: Find resultant}$

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

$=\sqrt{{\left[R(1+\frac{1}{\sqrt{2}})\right]}^2+{\left[R(1+\frac{1}{\sqrt{2}})\right]}^2+0}$

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

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

Hence , Option(B) is correct