Question

The resultant of the three vectors $\overrightarrow{OA},\overrightarrow{OB}$ and $\overrightarrow{OC}$ shown in figure is

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

Answer 1

Answer: (C)

If we assume horizontal and vertical as $X$ and $Y$ axes, then
$\overrightarrow{OA}=r\hat{j},\overrightarrow{OC}=r\hat{i}$
$\overrightarrow{OB}=r\cos 45^{\circ }\hat{i}+r\sin 45^{\circ }\hat{j}$
$\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}=\left(r+\frac{r}{\sqrt{2}}\right)\hat{i}+\left(r+\frac{r}{\sqrt{2}}\right)\hat{j}$
$|\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}|=\sqrt{{\left(r+\frac{r}{\sqrt{2}}\right)}^2+{\left(r+\frac{r}{\sqrt{2}}\right)}^2}$
$=\sqrt{r^2{\left(\frac{\sqrt{2}+1}{\sqrt{2}}\right)}^2\times 2}=r(1+\sqrt{2})$