Question

The resultant of the three vectors $\overrightarrow{O A}, \overrightarrow{O B}$ and $\overrightarrow{O C}$ 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
$\overrightarrow{O A}=r \hat{j}, \overrightarrow{O C}=r \hat{i}$
$\overrightarrow{O B}=r \cos 45^{\circ} \hat{i}+r \sin 45^{\circ} \hat{j}$
$\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}=\left(r+\frac{r}{\sqrt{2}}\right) \hat{i}+\left(r+\frac{r}{\sqrt{2}}\right) \hat{j}$
$|\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}|=\sqrt{\left(r+\frac{r}{\sqrt{2}}\right)^{2}+\left(r+\frac{r}{\sqrt{2}}\right)^{2}}$
$=\sqrt{r^{2}\left(\frac{\sqrt{2}+1}{\sqrt{2}}\right)^{2} \times 2}=r(1+\sqrt{2})$