Question

A sail boat sails 2 km east, then 4 $\sqrt{2}$ km south - east and then an additional displacement in an unknown direction. Its final position is 5 km directly east of the starting point. If the direction of the third part of the journey is at an angle $ta{n}^{-1}-n$, find n.

Answer 1

Let ${\overrightarrow{r}}_1$, ${\overrightarrow{r}}_2$ and ${\overrightarrow{r}}_3$ be the displacement components and $\overrightarrow{r}$ be the resultant displacement of the boat.
${\overrightarrow{r}}_1=2\hat{i}$
${\overrightarrow{r}}_2$=$(4\sqrt{2}cos{45}^o\hat{i}-4\sqrt{2}sin{45}^o\hat{j})$
$=4\hat{i}-4\hat{j}$
${\overrightarrow{r}}_3$=?
$\overrightarrow{r}=5\hat{i}$
$\overrightarrow{r}={\overrightarrow{r}}_1+{\overrightarrow{r}}_2+{\overrightarrow{r}}_3$
${\overrightarrow{r}}_3=\overrightarrow{r}-{\overrightarrow{r}}_1-{\overrightarrow{r}}_2$
$=5\hat{i}-2\hat{i}-4\hat{i}+5\hat{j}$
$=-\hat{i}+4\hat{j}$
$\therefore$ the direction of ${\overrightarrow{r}}_3$ with x - axis is
$\alpha =ta{n}^{-1}\left(\frac{4}{1}\right)$
$\Rightarrow n=4$